If someone wants to launch MVP and want to onboard first 10-20-50 customers then how much testing should one do in its application? Just manual testing and unit testing is enough? Or should it include integration, E2E testing as well?

Manual testing should be enough

I see honey badger is $40. Is basic free plan sufficient!?

Honeybadger has a free plan as well, start with that - upgrade when you hit the limit.