This grammar is $LL(1)$. A and B have single - single productions, so no need to even check for A and B.
And regarding the variable S, $FIRST(AaAb)$ ∩ $FIRST(BbBa) = phi$
Hence, this grammar is LL(1).
No, need to even check for LL(1) properties, if you look at the grammar, there are only two strings in it, 'ab' and 'ba', one string starts with 'a' and second string starts with 'b', hence no ambiguity, by looking at the first input element parser can know, that which production to choose.