Not in this context. The english sentence appears to mean "I do not count often (so I am not very good at counting, and don't know most numbers), but I know (the number) one".
Glossing "a lot" as "many" would mean you are (avoiding) counting a large number of things, and would make nonsense of the "but I know 1!" part.
In this Duo sentence, you could use "much", "often", or "frequently" for "a lot", but not "many". "Many" would only work for "a lot" if you are referring to the number, and is not an appropriate substitute for "a lot" when referring to the frequency.
(but that's just from my knowledge of english - I don't know whether كثيرا has same variety of uses that a lot does. And I know that expecting a Duo sentence to avoid nonsense may be unrealistic. Maybe that's what the author was doing for?)
Since „the“ was among the possibles selections, I wrote „I do not count a lot but I know THE 1“. Would someone be so kind and explain me why it is wrong, thanks a lot!
I'd guess because the arabic sentence does not use the definite article? I've observed that arabic uses "ال" in some instances where the english translation does not use "the", but I don't recall any instances where english includes "the" when the arabic translation does not use "ال". (I'm still learning, and may easily be wrong about this)