While there's nothing grammatically wrong with "I am invited to the party", and I would understand what you intended to say, it's not the way a native speaker would phrase this. If I was discussing my invitation to a party that hasn't happened yet, I would say:
I've been invited to the party.
(I have sounds far too formal, so I've gone with the contraction I've here.)
As for your second example:
I was invited to the party.
I think you're correct that no implication of time is made here. The invitation clearly happened in the past, but the party could have been a week or a year ago, or later tonight, or two weeks from now (or even now!) Some possibilities:
I was invited to Mark's birthday party last year, but then they cancelled it at the last minute!
I was invited to the party next week, but I've got the kids that weekend so I might not be able to go.
Why do I have to leave? I was invited to this party! I have the invitation right here, see?
So while the invitation took place in the past, in this case I would say that the actual party could be at any time; context will make that clear.