Based on the information you've provided, you stand a fair chance of seeing a "buy up for cash" upgrade offer right now if you view your reservation on united.com. The price should be similar if you see a buy-up offered at the kiosk or at online check-in time. If there are unsold J seats and no one on the upgrade list, you may see an even lower price at check-in time only (but not "tens of dollars", probably, order of magnitude, $400–$1000).

It is possible for UA to oversell the coach cabin and need to move people to a higher class of service to accommodate every sold seat. This is called an "operational upgrade". No one has documented the current order of priority for op-ups. If I had to guess, I would expect to see op-ups prioritized by status, then fare class, then last name.