I'm not a jackpot/spin and go player, so I can't speak first hand, but I've heard from very good spin and go
players that you want 300-500 BIs to play spin and gos comfortably. That would mean a bankroll of $600-$1000 to play the $2 games. That's what a professional player would use, if you're a recreational player you can probably get away with 150-200 BIs, so maybe $300-$400?
You may think that these recommendations are very high, which they are, but the swings in jackpot games are crazy. 100 BI downswings aren't unheard of.