How do the prediction intervals for ARIMA models compare with ARIMA-GARCH models? Should we expect the prediction intervals for one to be narrower/wider than the other?
No, we would not generally expect narrower or wider intervals on average. We would expect that the intervals (for a fixed-length forecast horizon) would have varying width over time if produced by an ARIMA-GARCH model but constant width if produced by an ARIMA model. This is because ARIMA model assumes the error variance (conditional on past information) is constant over time while ARIMA-GARCH takes it to be time varying.