When describing an action by two persons and that action has a negative outcome or nature, is it correct to name oneself first? As in I and he played poorly.
Placing "I" before another person is unconventional, see this answer on El&U which confirms.
It's nearly always: My husband/wife and I and not vice versa. There's nothing wrong or ungrammatical about your suggested order I and he played badly/poorly but it will sound "awkward" and unidiomatic to native speakers.
If you want to be polite and suggest that yours was the greater responsibility (for losing the match?) then you could say
Bob and I played poorly but I was [by far] the worst player.