Can a physician tell if a person has arthritis without taking xrays?

Yes. There are some forms of arthritis such as rheumatoid arthritis in which the diagnosis is made using a blood test. The diagnosis of osteoarthritis, which is a wear and tear arthritis, is made on an xray.
Yes. There are many different types of "arthritis" and many of them can be diagnosed without x-rays although they are not infrequently used even then to rule out other conditions.