Does the budget exclude all filling equipment such as pumps, tanks, compressors? That is, a PCP rifle under $1000, not including optics?
Good quality PCP air rifles are usually more than $1000, but there are many PCPs within the budget limit. As a general rule of thumb, you get what you pay for. Caliber choices in that price range will likely be .177 or .22 cal. If you haven't done so already, google Airgun Source and check out the many models they have.