If I caught the bus yesterday, I would be punctual.

Yes, it doesn't make much sense to me. What is it supposed to mean?

If I had caught the bus yesterday, I would have arrived punctually.

"If someone did something (simple past), someone would be (adjective)."
Assuming that "would be (adjective)" also happened in the past, is this construction possible? If yes, any examples?

Matthew Waiis this construction possible? ... examples?

It's probably not the most common construction, but it's possible.

If you arrived there within the next hour, the museum would still be open.
If I ran all the way through that park, I would be very tired.
If Laura missed the bus, she would be late.

CJ

I believe this is one of those " mistranslated" concepts that usually happen between people of certain Asian dialects and English construction.
If I understand your thought, this may be what you meant:

If I hadn't missed the bus (that I normally take), I would have been / arrived on time.