One can consider torsors (principal bundles) within any sufficiently nice category and with respect to descent with respect to given Grothendieck topology and given fibered category over the ground site. See for example
- Tomasz Brzeziński, On synthetic interpretation of quantum principal bundles, AJSE D - Mathematics 35(1D): 13-27, 2010 arxiv:0912.0213.
where in the main motivational part the codomain fibration and regular epimorphism topology are implicitly used. For the fibered category of modules (viewed as quasicoherent sheaves) over noncommutative rings, the faithfully flat Hopf-Galois extensions are the answer provided we accept Hopf coactions as dual representations of group actions. The tensor product is not a monoidal product in the category of associative algebras so this is a bit of a problem. Next thing is that one needs to consider nonaffine objects, if one is in algebraic framework, what can allow for more general concept of noncommutative principal bundles over the covers by noncommutative localizations which are analogues of covers in a Grothendieck topology. This kind of noncommutative principal bundles were skecthed in my articles
Z. Škoda, Localizations for construction of quantum coset spaces, math.QA/0301090, Banach Center Publ. 61, pp. 265--298, Warszawa 2003;
Z. Škoda, Coherent states for Hopf algebras, Letters in Mathematical Physics 81, N.1, pp. 1-17, July 2007. (earlier arXiv version: math.QA/0303357),
based on general picture of actions in noncommutative algebraic geometry as explained in the newer survey
- Z. Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183--202, arXiv:0811.4770.
See also the nlab:noncommutative principal bundle. I will hopefully release 2 more articles in this direction within next month or two.
Yes, for the spaces of sections of associated bundles with structure Hopf algebra one uses the cotensor product construction in the affine case; this is well known in the literature; the recipe can also be globalized by gluing along localizations. However this does not give the total spaces of associated bundles (in the category of noncommutative spaces) in satisfactory way in general, but only the spaces of sections.