Is there an algorithm that's "like" cross-validation for approximation algorithms of NP-hard problems?
Crossvalidation is used to test the difference in behaviour between training and test data. There's no equivalent notion for approximations. However, for many problems it's possible (often via an LP relaxation) to get a good lower bound for a problem. This can be used to test the quality of a given approximation algorithm in practice. 

For a good example of this, you should look at David Johnson's survey of experimental work on the travelling salesman problem.