EDC Tower Controller reported an arriving aircraft made an incorrect position report and flew across the departure end of the active runway without notifying ATC. The reporter stated there is no radar display at the facility and Controllers cannot see most of the aircraft until they are within close proximity.

Narrative

Pilot was flying VFR; checked on frequency to the east of the airport. I gave him instructions to enter a left downwind at midfield for Runway 13 and advised he was not in sight. He read back the instructions. He later keys up and asks if he's supposed to make a right turn to enter the left downwind. I told him if he was coming from the east at midfield like instructed; he was in fact supposed to make a right turn to enter the left downwind.I then noticed an aircraft flying over the departure end of my runway from south to north and verified with the pilot if that was him. He confirmed and said he no longer wanted to land and wanted to go on course to his destination. This whole ordeal was very unsafe and if I had traffic he would have been a very serious problem. If we had a radar display at this facility this whole situation could have been avoided without any sort of incident or confusion. We cannot see 90% of aircraft until they're about 1 - 2 miles away.This is unacceptable on the pilot's part as well as he should report the correct direction he is coming from. Please do something about this before something seriously bad happens. Get us a radar display; take some pilot certifications; and update our Management when actions are taken so we know something is being done. We do not have access to Cedar here as we are a new contract Tower. We don't have access to the FAA network to properly put in deviations. This is the 3rd report I have made in the last few months or so.