I have now found my firts Dragon Priest Drahgot.
I'm on level 16 and was able to clear my way to Drahgot, but there's absolutely nothing i can do to kill him.
He shoots a coulple of fireballs and i'm dead.

Even a battle axe with 41 damage (+cold damage) does so little damage to Drahgot that you really can't see it.
What kind of weapon or spell do i need to kill these dragon priests?

Unfortunately smithing isn't a good option to me, because i have just the basic skill and don't own any steel weapons.
I cannot use smithing to make my dwarven weapons better and i'm really not interested to create a pile of steel daggers
to improve the skill. I have to say that i don't like cheating, but if i cheat somewhere it is to increase my smithing skill a lot.
It is really stupid that you cannot buy smithing as a service.

Lower the difficulty. Gain some levels. Nothing else you can really do, since smithing is "cheating". I had trouble with this guy in full legendary daedric with 2h, only reason I survived was because I FUS RO DAH'd where he was easier to deal with.

I found fire and kiting extremely useful against merokai, not sure if it would work on this one though since i haven't found him yet and he sounds much more powerful.

Is this the dragon priest in Bleak Falls Barrow? If so, run down to the small stream and stand in it, then shoot him with arrows. If not, try to look for some area in the room where you can hide behind or stand on so you are above him, or run down a corridor. Sometimes they cannot follow you, then shoot him with arrows or spells.

There is no priest in Bleak Falls Barrow last I checked.

To the OP, do you mean Raghot in Forelhost? If so, I had some real problems with him too. After some trial and error, the way I finally handled him was to kite his minions (but not the priest himself) into the hallway leading to his chamber. After killing them, I hit the Ritual Stone power and let his own minions go to town on him. He had a sliver of health left by the time he killed them off again, so it was easy to finish him at that point.

Lord or Atronach stone, magic resistance any other way you can find it. (shield of solutude is quick to get)

the best tactic against dragon priests is to bash them in to a corner or a wall so they cant run away, and then alternate bashes with attacks to keep them staggered all the time, and its also a good idea to have some potions and cook vegetable soup so u dont run out of stamina, aand just lvl up a bit they are high lvl enemies with high lvl rewards

If you're being easily beaten, then you're too low a level. Enemies don't scale with your level as simplistically as in Oblivion; enemies like Dragon Priests are naturally high level, and will kick your [censored] until you get stronger. They're generally not meant to be taken on at a level as low as 16.

Obviously it's possible to kill them at that level, but I'd say they're designed more for the 25-30 range.