Different countries have different rules. In general an IR test done on a single will not be valid for multis (except non-asymmetric types eg C336/337. If you're authorised to fly one VFR then a S/E IR can cover it. Oz allows that, not sure about UK). An IR done on a twin usually gives S/E IR coverage as well. Perhaps not under JAR?
If you're tested on a single then you'll have to do another IR test on a multi to get multi-IR privileges.
As for cost, lengths of string come to mind. How much of the course would be in a synthetic trainer/sim/FNPT device, would it be a twin sim and at what rate vs. how much flying is done in a single and/or twin aircraft at what rate/approach & landing fees etc.?