Learn Finnish: Olin vähällä pudottaa lompakkoni.

The phrase olla vähällä means "to almost do something" or "to be on the verge of doing something". It is always followed by a verb in its 1st infinitive (the dictionary form), which is pudottaa here. Literally, it translates to something like "I was at a little to drop".

Could I just say Melkein pudotin lompakkoni instead?

Yes, that is completely correct and means the same thing. However, using olla vähällä + 1st infinitive is a very natural and common idiom in Finnish that strongly emphasizes how close the event came to happening. Notice that with melkein, the main verb conjugates (pudotin), but with olla vähällä, only the verb olla conjugates (olin) while the action verb stays in the infinitive.

Why doesn't the object lompakkoni have an -n at the end if it's a complete action?

You are right that the total object for a complete action normally takes the genitive-accusative -n ending, which would make it lompakon. However, there is a rule in Finnish: when you attach a possessive suffix like -ni (my) to a word in the genitive case, the -n disappears. Therefore, lompakon + -ni simply becomes lompakkoni.

I often mix up pudottaa and pudota. Why is it pudottaa here?