I don't often use taxis, but that's how we got from the port to the airport in Barcelona. I don't remember much about it, so it must have been pretty painless.
Our flight left early in the afternoon. Around 12:30, if memory serves. We were at the airport waaaay early. This was a few years back, so I don't remember all the details, but I do remember sitting in the check-in area for ages and watching masses of people pay to have huge amounts of plastic wrapped around their suitcases. It was the first time I'd seen it, and it seemed to be very popular in Barcelona. I was (still am) totally baffled over why anyone would want to do that, and disturbed by the utter wastefulness of it. But I digress. We were sitting in the check-in area because we were so early that no gate had been assigned, so we had no place to go. Possibly check-in hadn't even opened for our flight, so we couldn't drop off our bags. Can't remember for sure. Unless something drastic happens to delay your disembarkation, you'll have plenty of time.
International terminal. I can't imagine why United would have a counter in a domestic terminal at an EU airport. As a US airline, they won't be allowed to operate flights that both originate and terminate in the EU. That said, you can probably just tell your taxi driver that you're flying with United, and (s)he will know where to take you.