It was never called the "Inner Ring Road", perhaps because it went through the area known in Bristol as "the Centre" rather than around it. (Arguably, Bristol has three centres!) The number was never displayed on signs and is hard to find on maps, but I think it was A4044. Starting from the Centre and going clockwise, it ran along St Augustine's Parade, Lewins Mead, the Haymarket, St James Barton roundabout, Bond Street, then via an underpass to Temple Way, then across a notorious old single-lane flyover (supposedly a temporary structure!) into Redcliffe Way. Redcliffe Way ran over Redcliffe Bridge and then, incredibly, diagonally across the Georgian Queen Square, before continuing on to the Centre again. (Anticlockwise the route was slightly different in places, and didn't involve the flyover.)
The section across Queen Square was of course a monstrosity, and was closed to traffic a long time ago, traffic being diverted along the Grove and Prince Street (except for minibuses which ran round the perimeter of the square). But for many years the carriageway, complete with road marking, lay there empty as a bizarre "ghost road" across the middle of Queen Square, again because the closure was supposedly a temporary one!
A few years ago (I'm sorry I don't have exact dates) it was decided that Redcliffe Way should no longer be a through route, and the so-called temporary flyover was finally taken down. It's no longer possible to travel directly from Temple Way into Redcliffe Way, though you can still do so in the opposite direction. As for the road across Queen Square, it's been restored to its rightful status as a footpath. The minibuses no longer run round the square, and the road surface is currently being converted back to cobbles. Soon there will be no indication that it was ever a through route.
Temple Way/Bond Street is still a busy through route connecting the A4 to the M32 (and A38). At the southern end the sign says nothing but "(M32, M4, M5) (A420 )" a road sign with no destinations at all! The section between St James Barton and the Centre is also very busy. To get from Temple Meads to the Centre you're now directed along Victoria Street, Bristol Bridge and Baldwin Street, which was always a popular route anyway, and is currently numbered as the B4053 (it's presumably the original A4).
There's also a Bristol Outer Ring Road page.