When you say there is "power" at the light fixture, how are you measuring it? If you are using a digital multimeter and you are getting somewhere around 50 volts it's a phantom reading and you need to be using a solenoid type tester or introduce a source of resistance into your digital measurements (like a low wattage light bulb). You'll find the measurement drops to zero. Ditto for a non-contact type tester.

The nearest receptacle may not be the source of power for the light circuit. Have you measured voltage at the wall switch for this fixture (assuming it is switched)? If the light fixture does have a wall switch, is it controlled by a switch at a single location or by two or more switches (3 or 4-way circuit)?

Again, you're problems at the wall receptacle are likely the same phantom. The receptacle works fine if you plug in, say, a lamp. Is that correct?

I only had a tester(screwdriver with a light). It didn't light up at switch but it light's up at the wires from the bulb. Should i buy a voltmeter or somenthing?