"Other times" can be used with or without the preposition "at," often interchangeably.
The phrase needs to fit comfortably in the sentence. Of course, there will be
a reference to an earlier event For example:
Sometimes he would arrive late; other
times he would not arrive
at all.
At certain times he would arrive late; at other times he would
not arrive at all.
You can get your clue for which phrase works better
by connecting it to the previous reference.
The two sentences above are both fine. "At" or its absence is parallel
with the construction in the first clause. However, you could also add
"at" to the first sentence and omit it in the second sentence.
"Other times" seems a bit more descriptive of the event, while "at other
times" focuses on noting the event in a time frame. In addition, "at other
times" seems to have a slightly more formal register than "other times" does."