Should there be much pain after getting stitches in your palm?

There will be some. But "much" pain? Pain is relative to injury and the tolerance of an individual for that pain. And in a situation where sutures are used to close a wound, the sutures themselves are a source of some pain. A physician is the best person qualified to assist in the control of this kind of pain, and if it is simply "too much" pain, a consult should be obtained ASAP and something given (prescribed) for it. But you can expect some moderate discomfort (pain) after something like is described. If you can hang in there, use some over the counter (OTC) medication to dull the hurt. Consult your pharmacist for OTC meds, because they are able to offer suggestions and are marvelously equipped to point out a specific product you could use. As for dosage, ask for directions or consult those on the package, and operate by those guidelines.