Yes, a selfish person can be unselfish, because the way a person acts in most situations it not necessarily how they act in any one situation. Also, a person can act selfishly and unselfishly at the same time. For example, if a person feels good when they give money away, they can selfishly give to make themselves feel better.
Evolution has the animal kingdom kill and maim for survival. Survival of the fittest is ingrained in our mindset. We have to be self centred to survive, and though we have gone beyond the dark ages we still have those primal urges intact. That is, the urge to be the survivor, the dominant one. When push comes to shove most would do anything to live, to mate the partner they want, to get the spoils, wealth ect.
To be unselfish by choice is to break away from our primal desire, our desire to look after ourselves foremost which is something we all have by default. Everyone is selfish until the choices we make break it.
Only a very deeply selfish person can be truly unselfish. It may look like a paradox but the first basic thing about being selfish is to be self-centered. The second is to always look for ones own blissfulness. If your self-centered you'll be selfish whatever you do because you are doing it for you, you enjoy it and you feel happy.
A self centered person is always seeking their happiness. You cannot be happy if all those around you are unhappy as man is not an island or made of stone. You are very sensitive to our surroundings, very delicate and a miserable atmosphere will affect you. Therefore the more you seek your own happiness, the more you'll help others to be happy.
An unselfish person is always serious, deep down ill and in anguish. They are fighting to be unselfish when that is going against nature. Only selfish people can be unselfish. When you have happiness inside you can share it. To share you must in the first place have it.
A happy person belongs to himself. Rooted in themselves. An unhappy person needs to belong to something, some group because they have no roots in themselves, don't belong. That creates anxiety as they should belong so they create a substitute belonging. But if you are rooted in a crowd you belong to an impasse; from there no further growth is possible.
I'm assuming this question isn't supposed to be taken too literally, since anyone can do something unselfish even though they're usually selfish.
The main point I'm trying to make though is that If a person does something out of purely selfish reasons, then whatever they do is still selfish, even though it might appear to be unselfish to the people they're doing something for. So if someone gets enjoyment from helping someone in need and they're doing it for the selfish reason of feeling good about themselves, then they're actually still selfish.