I can't say i know exactly what you are going through, my older brother and I never got along, kids or adults. The question you posed in the title Should I or Should I not is easy, you should. I speak from experiance, if you let this stuff seep and fester in your relationship you'll regret it.
My advice isn't much different then anyone elses you've had so far. Appologize for the fight that occured, make sure he knows that you still love him and tell him you want things to be better between the two of you again. If he gets angry or hostile don't respond in kind, don't ignore the fact that he's angry but don't respond out of anger. If things don't go well the first time then tell him that you'll always be there for him if he needs to talk, let things settle for a bit then try again. Always be as positive as you can be, if he insults you accept it and move on, if he threatens you don't show fear, and if he attacks you walk away, but always make sure he knows you still care.
Best Wishes,
Nightowl88.