(I'll assume that in a general model category C, Cyl(X) really means: a factorization of $A\to X$ into a cofibration $A\to \mathrm{Cyl}(X)$ followed by a weak equivalence $\mathrm{Cyl}(X)\to X$.)
A sufficient condition on objects for the map in question 1 to be a weak equivalence, is that the objects X,Y,A be cofibrant. (The fact you want to use is the statement due to Reedy (which you can find at the start of the chapter on Proper Model Categories in Hirschhorn's book), that a pushout of a weak equivalence between cofibrant objects along a cofibration is a weak equivalence.)
A sufficient condition on C for the map in question 1 to be a weak equivalence, is that the model category be left proper.
It's an interesting fact that in Top, the this also works if Cyl(X) denotes the "classical" mapping cylinder construction (X union a cylinder on A), which isn't necessarily a cofibration in the Quillen model structure. The slickest proof is to use the "excisive triad theorem", as proved by May in A Consise Course in Algebraic Topology, p.79. This also leads to a proof that Top is left proper.