If I play Transmute Artifact, do I have to put mana into my mana pool before Transmute Artifact resolves, or can I tap my lands during Transmute artifact resolving?
You can activate mana abilities whenever you are asked to pay a cost, including during the resolution of a spell or ability. You can, but don't have to activate them beforehand.
605.3. Activating an activated mana ability follows the rules for activating any other activated ability (see rule 602.2), with the following exceptions:
605.3a A player may activate an activated mana ability whenever they have priority, whenever they are casting a spell or activating an ability that requires a mana payment, or whenever a rule or effect asks for a mana payment, even if it’s in the middle of casting or resolving a spell or activating or resolving an ability.