I’m not an “expert”, but to me 13x13 differs from 19x19 especially in how players should manage the middlegame and the late opening stage.
The first steps of the opening are quite similar: take corners, then take enclosures/corner approaches, then extend to the side by 4-5 spaces (if there are no bigger moves on the board).
In 13x13 games I don’t see enclosures very often, as usually corner approaches are also extensions from one’s corner, so approaches seem to be more efficient.
Especially approaches to opponent’s 3-4 stones facing one’s 4-4 stones are really good, as the 4-4 usually stands exactly “in place” for most josekis, so if you choose to approach a 3-4 with a variation that ends in gote on 19x19, on 13x13 you end in sente instead (because your 4-4 is already there).
Also, be careful when you choose a joseki: some 19x19 joseki are good in certain situations, but given the smaller board, situations are different. 
I don’t think I can give you any other advice, as I don’t usually play 13x13. And I don’t know if there’s some go literature about 13x13 around…