Well, first of all, the word you want is "intercollege", not "intercolleges" — the plural is unnecessary. If you want to sound even more professional, go with "intercollegiate", a bigger, fancier word.
"1st Prize" implies that you literally won a prize - in a situation like this, you're probably going to want to highlight the actual victory, rather than the prize associated with it.
"1st Rank" is a strange phrase that I've never seen before. I would avoid using it. Technically speaking, it is not incorrect, but I think it implies that there is a somewhat finite nature to a rank. In general, the word "rank" is more commonly used as a verb than a noun, except when referring to military rank. One could be a "high-ranking officer", for example, or one could have "ranked well in that competition".
For the first sentence, the award is for "Best Performance". The word "best" is a superlative, so you could simply say:
Award for "Best Performance" in State Drama Competition.
It would be slightly redundant to state that you scored highest at being the best.
For the second sentence, I would use the phrase "First place" (spelled out, rather than "1st place", if it is intended to be used formally), like this:
First Place in intercollegiate ‘Antyakshari’ competition.
Those are acceptable answers if you are being asked about awards or honors — however, if you want to place more focus on your achievements, or if you are being asked specifically about your achievements, I would reword them so that they are lists of verbs, rather than lists of nouns — this is a matter of taste, and so some might disagree. You do not have to word it like this.
I would write:
- Earned award for "Best Performance" in State Drama Competition.
- Placed First in intercollegiate ‘Antyakshari’ competition.
Unfortunately, there is no real English equivalent to "Antyakshari" — however, there is no shame in using a foreign word. You could probably say "Antyakshari singing competition", to indicate the basic nature of the competition, but that would perhaps take away from the mental aspect of the game. It's a tough call, but being multilingual is always a plus. English is a language of borrowed words - do not be afraid to add one if there is not already a word to take its place.