I also would've said "Sed mia kato ankaŭ manĝas muŝojn." The rule is that you put "ankaŭ" before the word it refers to, and therefore is never placed at the end of sentences. If it was "you and me too" in English, it would be "vi kaj ankaŭ mi" in Esperanto.
I think that "Sed ankaŭ mia kato manĝas muŝojn." could work too, because it would mean "my cat, too, eats flies."
English makes the distinction by emphasizing the word "also" refers to while speaking, ("I also play" vs "I also play") but Esperanto uses word order.
Is there any way to express the idea of present continuous in Esperanto? I know that "La kato mangas musojn" can be translated both to "the cat eats mice" and "the cat is eating mice" but I'm wondering if there is a way ti specifically say that something is happening in that particular moment as opposed to something that normally happens. Somerhing like " La kato mangas musojn [a word that means 'right now']"