This the old Severn bridge. It has two walk ways. From chepstow you can park in the Thornwell area and walk down behind the Tesco express to the walkway. Very relaxed code of practice is walking on the left walkway with the flow of traffic East, if on a push bike use the right walkway against the flow of West bound traffic. If you have children be warned, it is very easy for kids to get onto the live carriageway with only 2 low steel ropes devising the traffic from the walkway. Also the guard railings are not up to modern standard and are only about 4 foot tall. If walking keep to one side, advise close to railing, as cyclists and motorcycles use these walkways, many visitors gave been struck by cyclists, when walking and not aware what is around them. Also service vans use these walkways.
The bridge surface is a painted coating and in places bare steel plates, so it can be slippery and the surface is uneven. When you set off in Chepstow the weather and temp may be warmer calmer and dryer than up on the bridge.
The river Severn has a weather system all of its own. The bridge platform is shaped like an aircraft wing , so the wind comes up and over the deck and can be quite dangerous. At the other side of the bridge , ( 2 bridges , Wye bridge and Severn Bridge )is Aust. If you want a pint in Aust village I advise you walk over using the right walkway as this will bring you into Aust, however you will have to cross, the stupidly busy A403.
There is a services and large travellers site over at Aust but it is very poor and very expensive. Do not, and I mean Do Not leave your car in the services and walk over the bridge, Not saying it is down to one particular group of people but it is a well known hot spot for fuel theft and damage.
You can park on the old passage road, which is where the ferry used to operate from before the bridge was built.
Please note the new M4 bridge has no walkways.
This is a moderate walk of about 6 miles, Chepstow to Aust and back and well worth the effort. Best views down river towards the M4 bridge are on the right walkway.

Local place to eat in Chepstow if you park in Thornwell would be The Two Brewers, which is a chain style pub, with good value food and drink. There is a small Tesco express for essentials here too.