Can a UTI make you feel sleepy all day?

Sure. Any infection uses a lot more of your energy to combat it. Get plenty of rest and take your prescribed antibiotics and drink lots of fluids. Non-alcoholic of course!
Infections and sleep. Yes, an UTI can actually make you drowsy. The immunomodulators (immune system components) that are involved in fighting off infections also affect the brain with regards to sleep and wake. When you have an acute infection of any sort, these immunomodulators actually partially suppress the part of the brain that is responsible for keeping one awake & hence you feel drowsy.
General no. Anything that makes you feel bad can make you tired or sleepy. Are you running a fever? Without a fever, you will have discomfort from a UTI but should not expect to feel sleepy.