Usopp, Nami and Chopper in One Piece. Nami is partly justified because she lacks the insane superhuman strength and/or Devil Fruit powers a lot of her crewmates have, so she must rely on her brain, wits, and Clima-Tact to survive in battle. As she gets skilled with the Clima Tact (and no longer has a crappy party favor toy), she becomes more courageous. Chopper is fairly young, shy, and sheepish. He freaks out a lot and gets shook up in bad situations. Post time-skip, he's more in control.
Of course, Usopp's a funny case in that as long as everything is fine, he's a total Miles Gloriosus. But the instant there's the slightest hint of danger, and particularly when he's the only one who can do anything about it, he upgrades (downgrades?) to Cowardly Lion. When any significant danger presents itself, Usopp will panic, then proceed to kick ass despite shaking in his boots the whole time. As Sogeking, he tries to ham it up under the facade of a capable badass, but the disguise melts away as soon as real danger comes his way or he gets annoyed. After the time-skip, he gets a bit better, and resembles a real badass- though with some cowardly traits still very much intact.
Karoo from the Baroque Works Saga is a lesser example.
Shinji Ikari in Neon Genesis Evangelion. Normally a timid, withdrawn, and slightly neurotic person, he gets surprisingly confident when he pilots the Evangelion, but he quickly falls back to his usual behavior outside of it.
Mr Satan in Dragon Ball. And to a lesser extent Krillin.
Zig-zagged, especially during the Saiyan Saga, where he's just as likely to panic and cower in a corner at a crucial moment as rise to the occasion. His level of bravery seems directly proportional to how hard the villain is hitting his Berserk Button.
Jacuzzi Splot from Baccano!. The guy knocked over eighteen Mafia-owned speakeasies in one night by himself, with a single Tommy gun, crying hysterically in fear all the way. The week after, he helped save a train from two clashing terrorist groups; to do this, he did battle unarmed but for a single cherry bomb against someone with a shiv knife, a fist-gun and a flamethrower (which did turn out to be pretty convenient for lighting said bomb) on top of a moving train. He took a couple of bullets in the gut and a knife in the elbow before he tackled the other guy off the train and won. He was later half-carried to the station by his girlfriend, moaning in pain.
Kobanzame of the Kyoshin Poseidons is another example. A nervous nellie of a quarterback, Kobanzame thinks he's useless and leaves all the major decisions to Kakei. Yet he has perfect ball handling, and on short passes is a perfectly competent player. Reiji Marco, captain and quarterback/safety of the Hakushuu Dinosaurs is a villainous example—he's utterly terrified of getting hurt, yet continues to use himself as his own best piece, risking his neck to lead his team to victory.
Simon from Tengen Toppa Gurren Lagann is like this initially. Scared of all the things happening around him that nearly kill him, but if Kamina is in danger he opens up a can of badass at a moment's notice.
Closer from Hyde And Closer. Always run from battle, Hyde always had to bail out his ass, but never let his friends in danger, even if that means takes the beating all for himself.
Kamiya Maki of All Rounder Meguru first finds Aya huddled in the corner of a room at a kickboxing event. She darts off like a rabbit the moment Maki tries to talk to her, leaving Maki bemused to discover that Aya is actually her next opponent, and astonished to find that she's the most formidable opponent she's ever faced in a kickboxing match.
At one point, Kenichi is induced with a 'mental seed of fear' after a near-death defeat by one of his enemies. It took the efforts of one of his teachers who essentially invoked this trope when she did not intend to not cure him of his fear, but rather to teach him to weaponize it as a means of self defense in order to overcome its negative effects.
Kain Fuery in Fullmetal Alchemist. He's the smallest and youngest member of Roy Mustang's hand-chosen unit, and his main value is as the communications specialist. When the team gets split up and he's reassigned to the southern war front, he's absolutely terrified and only gets through it by shouting at himself that he's going to live. Later, he deserts in order to return to Central City and aid Mustang and Hawkeye with their coup.
And then we have Warrant Officer Falman, who was literally crying and shaking with fear as he defied Bradley's order to open the gate to Central palace. Captain Buccaneer called him on it.
Buccaneer: Oh come on now. Don't cry like a girl when you act like a man!
Kai Shiden of Mobile Suit Gundam with emphasis on "Lion". An abrasive jerk who perpetually points out the stupidity of the situations they're in and often seems about to panic when sent into combat, Kai's nevertheless a reliable solider who racks up a pretty impressive kill count over the course of the war. He might have easily ended up as one of the Earth Federation's most celebrated aces, if not for being the wingman of the White DevilAmuro Ray.
While Ranma Saotome of Ranma ½ is a Combat Pragmatist, he's generally pretty far away from anything that resembles cowardice... unless cats are involved, in which case a few seconds after being surprised by a housecat he is literally catatonic and will eventually lose his mind from the sheer terror until he starts thinking that he's a cat and starts kicking the ass of almost everything within eyeshot using the ultimate technique, the Neko-ken. However, when he was tossed into a situation that almost perfectly replicated his initial trauma (being tossed into a pit of hungry cats while wrapped in fish products), he managed to (sorta) maintain his composure until they brought out a tiger that was about as tall as he was, at which point he snapped. Later on, when facing a grandmaster that was casually kicking his ass and holding his manhood hostage, Ranma deliberately strapped a cat onto himself (or rather, herself) in order to pull out the Neko-ken and get his opponent to back down. That is to say, he deliberately drove himself insane with fear in order to win a fight.
Kasumi of King of Thorn. She believes she is weak and useless, having depended on her braver twin sister Shizuku for most of her life. Later she depends on Marco for protection and often needs him to rescue her. However, she's still able to put her fears aside long enough to save someone else's life. She even saves Marco a few times, and earns his respect for it.
Arata Hinohara from Arata Kangatari is understandably scared to fight and follow his destiny in the beginning of the series, but he follows through with completing the task at hand, anyway. As the story progresses, he gradually grows out of it.
Suigetsu from Naruto tried to run away from Team Taka as soon as he gets the chance, despite being made of water and completely impenetrable. Too bad non-combative Karin is faster than him and prevents him from getting away.
Attack on Titan has many of these, which is understandable considering what they are upagainst. Among the main characters, the following stand out.
Sasha Blouse is a Cloud Cuckoo Lander, and breaks down during her first battle. She even apologizes to a Titan she'd failed to kill, and agonizes over it later. But she's also managed to charge down a 50m wall to rescue an injured comrade, and faced down a Titan with nothing but a hatchet and a bow to save a child.
Bertolt Hoover readily admits to being an Extreme Doormat, and constantly stands there awkwardly with a slightly terrified expression. He's also ranked 3rd in their class, and is noted by their instructor to be one of the best at physical combat. He's also the Colossal Titan that smashed open Wall Maria and Wall Rose.
Any soldier that shows fear prior to combat. It's not uncommon to see the same person that was weeping hysterically prior to battle turn into a Hot-Blooded warrior when the chips are down. Unfortunately, since it's that kind of story, many of them still die horribly.