Stanhope from Journey's End
's a good example of a likeable mean character.
In that case it makes him sympathetic by attributing his Kick the Dog
moments to factors outside his control (drinking to cope with the stress of war, combined with a fear of his friends and family finding out what a wreck he's become) and making it clear he's quite a pleasant person otherwise with Pet The Dog Moments
A particular scene with Hibbert (a character who's been pretending to have a hard-to-disprove condition so he can be sent home) might be worth looking at. Standhope basically tests Hibbert by threatening to shoot him (arguing that it's better to be shot by "accident" than be executed for attempted desertion) and offers support when Hibbert asks him to do it
. IMO it works well because it allows him to Pet the Dog
and explain his own motivation/positive qualities (it makes it clear he values honour and shows what's going through his head by having talk about it with someone who's having similar problems). He also goes from mean (Paraphrased "Try to get out of this and I'll shoot you!") to understanding ("We're all going through the same thing.").
There's also the begining of the play, which has two characters talking about him; one of whom (who's a bit of a jerk and later shown to be at odds with Stanhope) gossips about his drinking, the other (who's a kindly Father to His Men
) defends him. A scene like that might help.
Of course that only works if you want
to have your character absolved of responsibility for their mean-ness.