In the quest guide on "priest in peril" it mentions no quests are needed. But isn't rune mysteries a requirement for it? THe character mentions having recently helped some wizards in wizard tower, so i think that would be a requirement. But i'm not sure.

You don't need to have done Priest in Peril, as far as I know.
You may need to if you wish to mine all of the Rune/Pure Essence that you need, but you can also buy it from another player/GE so you wouldn't need to have done the quest for that.

If I am correct, it is just a slight twist in the storyline, depending on what quests you have completed.

it's not needed, and the dialogue your talking about is probably a small miscalculation by jagex, seeing as they probably thought you would get the essence for the quest yourself instead of buying it.

It's not needed, but it's an advantage to mine the pure esscence to save you a load of money.