The nearest car park is the surface pay & display one on Glendower Street:
https://ibb.co/y0rPTMT
It's free from 6pm to 8am. At other times it costs £1.50 for 2 hours, £1.90 for 3 hours or £2.40 for 4 hours (which is the maximum stay during chargeable hours). If you happen to be there on a Sunday though, it's free for 2 hours or £1.00 all day. Payment is accepted by cash or card. (Monmouthshire Council don't seem to have signed up to any app-based parking scheme yet).
Some car parks (e.g. those in Whitstable) allow you to 'carry over' parking fees to the next day. e.g. if you pay for 2 hour's parking just half an hour before the day's chargeable period ends, you get the first hour and a half on the following day thrown in. Similarly, if you pay for 2 hours parking at midnight (when parking is free), the system will automatically give you the first two hours of chargeable parking in the morning. Such systems are great when you're staying overnight in the area, as you don't have to dash out of your hotel at breakfast-time to pay for parking until you want to use your car. I've no idea about whether or not Monmouthshire County Council car parks use such a system but you might be able to find out by calling 01633 644644.