In an environment where people work together on important projects, conflicts are bound to happen. Once there's tension between you and a co-worker, there's a temptation to become overly defensive with every type of interaction you have with that specific person.
Don't raise your hackles every time you pass this person in the hallway. Pick the issues that you simply can't compromise on and let other things slide. Being generous in compromises with your co-worker will make it easier to win on the issues you really do care about.
Absolutely avoid anticipating conflict. If you do, you may be feeding into a self-fulfilling prophecy in which tensions escalate higher and higher for no good reason. Try to approach interaction with this co-worker with a sense of objectivity, rather than building a sense of hostility or defensiveness beforehand.
Realize that trying to get the best of this person every time you interact is probably not a good strategy. Doing so is time consuming, emotionally draining and completely counterproductive.