how to use type application rule (T-TAPP)?
Type checking and the inference rules for type theory are not an algorithm. They do not tell you how to type check. They tell you what is allowed when you perform type checking.

The rule for application should be read as:


  If you can prove $\Gamma \vdash t_1 : \forall T . T_{12}$ (and we're not telling you how to do that, you'll have to figure it out) then $\Gamma \vdash t_1[T_2] : [X \to T_2]T_{12}$ for whatever $T_2$ you choose to use (and of course we do not know what it is that you're trying to do here, the choice of $T_2$ is entirely up to you).


So, the answer to your question is: you have to figure out how to find the value of $X$ and you have to worry about the subtitution. The rules are not meant to solve that problem. They are just rules, telling you what you may do.

The correct question that you should have asked is: "How do I make an algorithm that will be able to perform type-checking automatically?" This is an important question, and people put a lot of thought into it. I can give you an initial answer: try out all possibilities until you find one that works. But that's a lousy aglorithm and it's possible to do better.