We gotta meet up real quick. - the meeting is "real quick" or for a short amount of time. This would normally be something that takes place right now, but could be scheduled. It would be better to say "we need to meet real quick", but "we gotta meet real quick" would be ok in a really informal setting since "gotta" is not a proper word.
We need to meet up urgently. - Means that it is very important, but has no implication of exactly when or for how long. Although, soon would be implied since it's urgent. This sentence is poorly formed.