How to build a parser for syntactic analysis of regular expressions?
It is absolutely possible and you are on the right track.  You will need to make sure your grammar is %100 correct and probably LL(1), from there you can choose one of many algorithm examples that are out there or write your own.

See https://en.wikipedia.org/wiki/Top-down_parsing and https://en.wikipedia.org/wiki/LL_parser

Word of warning: The tricky part won't be the implementation but getting your grammar right.