1) The book by Sergey V. Matveev
Lectures on Algebraic Topology, Sergey V. Matveev (EMS Series of Lectures in Mathematics, 2006)
contains about 10 pages of hints and solutions to its exercises. That's not a bad ratio since the body of the book is only 82 pages long! Despite its terseness the book is amazingly nourishing: it contains Wirtinger's presentation of the fundamental group of a knot, higher homotopy groups, Poincaré duality, the Lefschetz fixed point theorem,... all starting from scratch. Of course not everything is proved but the numerous exercises prove that there is real meat to feed on in that short tome.
2) The book by T.W. Gamelin and Robert E. Greene
Introduction to Topology , Dover (2nd ed. 1999)
devotes its second half to algebraic topology (the first half being about general topology). There are 13 pages of solutions just for the exercises of the algebraic topology part.
Here also the material is more advanced than one could expect in an introductory book: higher homotopy groups, Jordan theorem, degree of maps ,... The price ? $9.15
3) The book by H. Sato
Algebraic Topology: An Intuitive Approach, AMS (1999)
is also a book starting from scratch and reaching fairly advanced subjects in 110 pages, while giving serious technical proofs on the way. The only book I know with solved exercises on spectral sequences!