According to director (and cast member) Jonathan Frakes, Drunken Troi is worth the price of admission alone.
Cochrane's initial reaction to the situation:
Cochrane: Hot damn, you're heroic! *laughs*
After Cochrane becomes convinced of the truth, Riker goes on to explain how the approaching aliens will detect his warp flight and "land right here."
Cochrane: Here? LaForge: Actually, over there.
A large portion of Zephram Cochrane's scenes, but especially:
Geordi:(engages in long spiel about Cochrane's magnificent standing in 24th century history, and the giant, inspirational statue of him that will stand right where he is standing now) Cochrane:(beat) ...I gotta take a leak. Geordi:Leak? I'm not detecting any leak. Cochrane: Don't you people in the 24th century ever pee?
Its also funny as part of Trek's subtle running gag of referring to the original Enterprise's comical lack of bathrooms.
There is something hilarious in Cochrane's motivation speech. Everyone is making him out to be this hero and visionary, and he's just in it for the money. But especially hilarious is when he explains that he really shouldn't be doing this:
Cochrane: I don't even like to fly!
Cochrane's exclamation of "SWEET JESUS!" when the Enterprise casts its shadow over the Phoenix.
Also, Cochrane's hilarious "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAHH" when the Phoenix finally breaks Warp Factor 1.
The Defiant Conn officer having a subtle worried glance when Worf states that 'Perhaps today is a good day to die', followed by the order to ram the Borg Cube.
When the helmsman tells him the Enterprise is arriving, Worf almost looks disappointed hearing that.
After the Enterprise is ordered to the Neutral Zone, the scene between Picard and Riker is brimming with Sarcasm Mode.
Riker: We finished our first sensor sweep of the Neutral Zone. (hands Picard a PADD) Picard:(reading) Oh, fascinating. Twenty particles of space dust per cubic meter, 52 ultraviolet radiation spikes, and a class-2 comet. Well, this is certainly worthy of our attention.
During the Dénouement, observe the Vulcan's reaction to Cochrane blasting Little Richard's "Hoobie Doobie". He takes a huge gulp of booze and wears a "What have I gotten myself into?!" look on his face.
When Data keeps asking questions of the Borg Queen, she asks - with perhaps a touch of exasperation - "Do you always talk this much?"