You can definitely paint it and that too by yourself if you are ready to put in little efforts. You can use epoxy paint for giving a new look to your old tub.

Use epoxy paint in the spray gun. Epoxy paint is a two-part paint that hardens to a shine. Apply two to three coats to the bathtub, using a left to right sweeping motion. Make sure the paint has plenty of time to dry between each application. You can also apply the paint with a paint brush, but it will take much longer and won't look as smooth.
You should top it up with an acrylic coat to finish it off.

It’s a messy job but worth the effort as you save a lot of money and the satisfaction from doing it yourself is immense. Don’t forget to cover the surfaces you do not want to paint by taping some paper or plastic over them. This will most likely include the faucet, hot and cold water knobs and the floor around the bathtub. All the best!