For 2 days I have a hard, painful bump on my back, the size of a fingerprint. It is maybe 5 CM tall. It's red, with no pus or skin lesion/rash.?

Lesion. All skin lesions like this should be seen by a clinician with possible referral to a specialist. It is impossible to diagnose these online.
Many possibilities. There are many types and causes of skin rashes. Several types of inflammatory conditions of the skin, allergic reactions, and sometimes bacterial, viral or fungal infections can cause rashes on the skin. Insect bites can also be the cause of a rash on the skin. Have it evaluated by a professional and get the appropriate treatment.