Short answer, yes a cheapie grinder will do OK for a plunger (in most cases). A better grinder will always make for a better coffee, but sometimes finances just dont allow for it.
Long answer, If you can stretch the budget, by all means buy a better grinder. However, if you can only afford up to the $100 mark, or you dont want to spend a great amount on a grinder (EG: $200+ is probably entry level for anything decent), youre probably not going to find anything that will last you through any significant part of the coffee journey. Better to make do with the cheapie and save up for a nice one later on.
As for storage. Dont grind and store. Ground coffee is best used within a few minutes. Youre better off getting an airtight container and storing whole beans away from direct sunlight and extreme temperatures.
Home roasting is great fun and can be started with a popper, a heatgun /bowl/ spoon, a heatgun and old breadmaker etc. GET INTO IT! Youll love the results.