There is no time limit, it seems. You could save up for the item the Fountain Faerie wants for years

Then use the strategy with the switch account thing...for normal quests.
I've played NQII all Faerie Festival Day, still no fountain

I'd paint my Jetsam Robot, though. If I could paint all my pets, I'd do:
Electric Jetsam, hazard_super, a robot Jetsam (what a 120k waste for the electric p/b)
Superior_Hazard, Silver Lupe, Mutant.
Radical_Prowler, Red Kougra, Tyrannian.
Might as well not paint Hazard_Hero, since he's labbing. Maybe one day the lab'll zap him MSP.....or alien aisha, or sponge...yeah.
When the Fountain faerie gives you a quest, can you only paint your active pet?