Student or Learner
why don't you take this opportunity that I am in town and ....
why don't you make good use of me being in town and ...
which one is the best, please?
First, as there are only two sentences, the question is which is better, not best. Best requires more than two choices.
To the first sentence, I might change it to "while I am in town" rather than "that."
To the second, [almost] every time you see a gerund (an -ing word) used like this, you need a possessive. It should be "of my being in town" for it is the being in town that you want to take advantage of.
Exceptions would be with something like this: Listen to me talking like this. You could actually put a comma after me: Listen to me, talking like this - thus making two phrases. But with your example above, "take advantage of me, being in town" is awkward. Or wrong.