[Experience: US consulting and academia]
In general I would say no, but it depends a little on the context.
Having a (mostly) empty drink in front of you can highlight the fact that you arrived first and indicate how early you arrived. If this is a situation where it would be embarrassing for you to have arrived significantly before the other person, don't do it (also, in such circumstances, don't arrive early).
If the meeting is specifically to get drinks, then it could be considered rude by some people. This is similar to how it's rude to begin eating before everyone is seated at a table. If it's a meeting that happens to be at a bar and there's no expectation of drinking as a group activity, I would think it wouldn't be considered rude.
I would avoid having more than one glass in front of you though. That can give off the impression that you've been waiting for a long time which can embarrass the person who has made you wait, even if they aren't officially "late".