In cash game poker 5BI downswings are very common, 10BI downswings happen regularly, and 20-30BI downswings happen occationally. So if you play 2NL, you should expect to lose 10$ fairly often, 20$ regularly and 40-60$ occationally.
Its not strictly nessesary to have that money deposited in your poker account, since it only takes seconds to top up your account with a credit card if needed. But for many of us it feels better, if the balance in the poker account dont undergo huge swings. And it can also be nice to keep poker money separated from our general budget. For those reasons its reasonable to deposit at least 60$ before starting to grind 2NL.
I think, you are also asking, when you will know, if you are actually beating 2NL? There is no simple answer to that. Try to play around with this calculator a bit. Typical variance numbers are 100 for 6-max and 70-80 for full ring.
https://www.primedope.com/poker-variance-calculator/
Personally I dont think, it is nessesary or benefitial to spend all that much time at 2NL. Maybe 20.000
hands is enough, before you start to challenge yourself against the slightly better players at 5NL. Its not a major issue, but there is some risk, that playing to much at 2NL will teach you some bad habits, which you then need to unlearn later.