Use "in the city centre". A city centre is usually an area which contains many buildings. "At" would suggest a very specific location - "at the very centre of the city".
"... destroyed by fire" (no article) works better for me. I find "... destroyed by a fire" a little awkward although it's not ungrammatical. "... destroyed in a fire" works well.
The original sentence needs commas:
The 200-year-old library, which was located in the city centre, was destroyed by fire last night.
Student or Learner