As much as I would too advice you to browse other questions about Rome for details, I can answer you with the following:
What shoud I see with highest priority?
If you are staying near the Termini station (main train station), you can walk via Nazionale or via Cavour from the station to Via dei Fori Imperiali, which is the road that stretches along the Coliseum and the (of course) Fori Imperiali. Given your timeframes, you are probably not going to find anything open (provided that you need an hour to get back to the hotel, change and have a shower). At the end of the road there's Piazza Venezia with another huge monument (Altare della Patria). From there you can go to a lot of places, but I would advice going to Piazza del Popolo, Piazza di Spagna, Piazza Navona, Fontana di Trevi (google up what you can find there).
Definitely a wise choice would be to spend a night around Trastevere (which you can get to by metro, or bus), explore its narrow streets, have some drinks and order to eat by an "Osteria".
Any not too expensive shops/ markets where I can buy something to drink/eat?
Supermarkets would be the best choice, but you can find a lot of "Pizzeria" where you can buy just a piece of pizza and soft drinks or even beer. There are also some ice-cream shops and fast-foods there, so I think you won't starve.
Bars and restaurants where I want to try pastas, pizzas and local food with not too high prize?
To be honest I avoid that area at all costs, when it comes to eating. A safe choice would be going in direction Trastevere, where you can even ask some people what you should eat. Pizza and fingerfood like Supplì are typically good and cheap, but really you should ask young people.
There is, now that I think about it, one place near Via Cavour where you can eat some pretty good pasta: La Carbonara. Goes without saying, you should try eating carbonara there.