"I thought that I bought beer."
Translation:Cheap mé gur cheannaigh mé beoir.
It’s not technically a past tense (since it’s a conjunction, and conjunctions don’t have tenses), but this gur is only used before a regular verb in the past tense.
Could you say something like "cheap mé go raibh beoir cheannaítear agam"
That’s somewhat scrambled in Irish. Which English sentence are you trying to express in Irish?
In the present tense you do indeed have the option of the one-word form for the first person singular: ceapaim. But in the past tense, you always have to use the form "Verb + mé" for the first person. "Cheapaim" doesn't exist.