What all can a valid approximation algorithm access?
Depends. Typically, with "approximation algorithms" people understand algorithms that run in polynomial time and return a possibly non-optimal solution (with a formal guarantee on the solution quality). In this definition, no, you can't spend exponential time solving an NP-hard problem.

On the other hand, nothing is forcing you to consider polynomial time algorithms that return non-optimal solutions. In fact, it is interesting to consider algorithms that take exponential time, and still return non-optimal solutions. See for instance [1, 2].



[1] Cygan, Marek, Lukasz Kowalik, Marcin Pilipczuk, and Mateusz Wykurz. "Exponential-time approximation of hard problems." arXiv preprint arXiv:0810.4934 (2008).

[2] Cygan, Marek, Łukasz Kowalik, and Mateusz Wykurz. "Exponential-time approximation of weighted set cover." Information Processing Letters 109, no. 16 (2009): 957-961.