"At the palace" and "in the palace" would both be translated as "w pałacu". "Na pałacu" sounds more like "on top of the palace", as in "on the roof".
Is it normal that the "L" in pałacu is pronounced as a Russian "L", instead of "U" as it normally is in other words? I mean, is it an exception only for this word, or is it just a mistake of the program?
Ł is generally pronounced like English "w" (word, world), always.
I was told by some Russian speakers that it's not how Russian Л is pronounced, although I heard Polish people from my parents' generation pronounce it like Ł...
I also wouldn't say that Ł is pronounced like U, it's rather the other way round sometimes, if U is preceded by a vowel it does sound like Ł (auto -> ałto, dinozaur -> dinozałr, etc.)