What are prerequisites (in mathematics and physics), that one should know about for getting into use of ideas from non-commutative geometry in physics?
I am not a fan of Alain Connes's ideas of non-commutative geometry, I prefer Michael Artin. Nor do I think he was Dixmier's best student, although Dixmier thought so. Nevertheless, to be fair, you should understand the physics of General Relativity and Quantum Field Theory first. I recommend Einstein's original papers still, and Yvonne Choquet-Bruhat's two books, Géométrie Différentielle et Systèmes Extérieurs, short and to the point and mathematically clear...obviously it is still commutative geometry, and her much longer, too long, Analysis, Manifolds, and Physics with two excellent co-authors. I also recommend Dirac's own short physics book on the subject. Then for QFT, Pierre Ramond is accessible and standard but very far from the point of view of Connes, which is why it would be good for you to read so you do not stay way too isolated. Also for QFT read the work of Irving Segal (sometimes with Roe Goodman) on relativistic quantum fields from the operator algebra point of view, but he is not a physicist so you should also look at Streater and Wightman PCT, Spin and Statistics, And All That... to see how physicists do operator algebra approaches to QFT.
If you do not yet even know Quantum Mechanics, I can recommend Sudbery's Quantum Mechanics and the Particles of Nature.
I assume from the way you phrase your question that you already are familiar with Connes's work itself. If not, you would need to prepare by studying Dixmier's book on von Neumann algebras or Guichardet or Dixmier, Les $C^*$-algebres et leurs représentations. Hope this helps.
Physical applications of noncommutative geometry rely mainly on the spectral triple formalism. This being a noncommuative version of Riemannian (spin) geometry, it would be advisable to know then first the commuative case (spin geometry itself)-Friedrich's book on Dirac operators is a good reference. This generalization uses lots of operation algebraic concepts. Varilly's books on NCG are recommended. The recent survey arXiv:1204.0328 is adressed to physicists.
The next level would be the book of Connes and Marcolli: "NCG, QFT and Motives " http://alainconnes.org/docs/bookwebfinal.pdf