Practice is #1, yes.
But... and I know I'll probably catch some flak for this... if you're good with protools (or whatever DAW) and things are a little off beat, you can zoom in and drag notes around, using cutting and pasting combined with fades/crossfades to extend the sustain of a note that had to be moved forward, and sort of a reverse process for moving a note back in time. Some engineers don't like it when people take their parts home for editing, but I for one appreciate a band that understands enough about pro tools that they can say, "hey, let me take my bass parts home and edit things a little and bring it back to you edited". Make sure you dont' overdo it.
Next, just make sure you're grooving with the drummer as much as possible for the whole session. Even focus on watching the head of the kickdrum and try and play in time with the way the head of the kick is moving in and out when the beater hits it. Make sure you can feel the bass (kinda sucks to record in just headphones), so you might want to consider baffling your amp or being on the other side of the glass from the drummer if your studio has that luxury. Ideally you'll be in the same room as the drummer so you can really feel the drums too. Make sure the band as a whole lets you and the drummer get good takes... the guitar and vocal stuff can be overdubbed later as long as the drum/bass foundation is solid.
Whatever happens, focus on the big picture. Simplify your parts if you have to. Nobody is going to listen to your record and say "man, I wish the bassist was playing 32nd notes instead of 16th's or 8ths there", but they will hear something that was too fast for your ability or the part that is best for the song. Lock in with the drummer, focusing on the groove of the whole song, and don't stop a take just cuz of a few notes. If there's good isolation on the bass rig, and assuming you're not tracking to tape, those parts can be edited to be more on beat, or overdubbed later.
Also, you might try moving to a lighter pick or play closer to the bridge or closer to the neck; see what's faster/easier and go with it. When I play with a pick I use Clayton bass picks. The triangle shape really rests in your hand very well, and the slightly rounded corners make the pick move across the strings more consistently than a guitar type pick. They are the best for this type of playing. Seriously, GET CLAYTON BASS PICKS. Focus on your up and down wrist/forearm motion ("alternate picking") and make sure that the up-and-down motion isn't interrupted by a certain line, and if it is and it's causing problems, change it or play it on a different string or position and see if that helps.
Finally, remember its punk, so try not to give so much of a f*ck and focus on being high energy. Hope this helps!
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