You can use the added by a Powerstone token on anything that isn't a nonartifact spell. This includes paying costs to activate abilities of both artifact and nonartifact permanents, paying ward costs, and so on.

Powerstone tokens are a kind of predefined token. Each one has the artifact subtype "Powerstone" and the ability ": Add . This mana can't be spent to cast a nonartifact spell."

Although all the cards in The Brothers' War that create Powerstone tokens create a tapped Powerstone token, entering the battlefield tapped isn't part of the token's definition. Notably, if you create a token that is a copy of a Powerstone token, the token copy won't enter the battlefield tapped.