Place to leave my luggage in Barcelona for one night and get it back early in the morning

According to the El Prat airport webpage, there are indeed left luggage facilities in both terminal 1 and 2. The former is open 24h whilst the latter is open from 0600 to 2200. The linked website carries information on directions and prices:

Left-luggage office

Luggage storage and management service located in the intermodal lobby of terminal T1 (floor P0), on the lowest level of the Plaza, and in the check-in lobby of terminal T2. The T1, open 24 hours, the T2, from 6.00 to 22.00 h. The prices are:

Normal luggage (suitcases or similar)

For a 2-hour period, €6/item of luggage; for each 24-hour period after the first 2 hours, €10/item of luggage.

Here are access maps for the Terminal 1 left luggage facility, and the Terminal 2 left luggage facility.

Another question: are there buses or trains around 5 a.m. that go to the airport from the city center?

Important note: check carefully what is the terminal you are flying from. It really matters!

There are local trains from Renfe running from the city center to the airport of El Prat de Llobregat. You can find the timetables in: http://www.renfe.com/EN/viajeros/cercanias/barcelona/index.html (in English).

What I would do is to get the train from "Passeig de Gràcia", which is nearby Plaça Catalunya. Its price is around 2 euros.

The first one passing by Passeig de Gràcia is at 5.08 and you arrive in Terminal 2 (the old one) at 5.30. From there you can get a shuttle bus (for free) that drives you to Terminal 1 (where most of the companies fly from). All together, you can make it to your terminal in around 40 minutes.

However, it may be better to get the Aerobús, which is a bus going directly to Terminal 1. You can get the timetables from http://www.aerobusbcn.com/en/timetable (in English). A single ticket is currently 5.90 euros. It is a bit more expensive but drives you to your terminal in around 20 minutes.