Depends on what the mod does and whether it modifies base classes.
1.2 introduced a new save format, jungle biome, and taller chunks. However, none of those should affect most simple mods. More complex mods might, though, but you can usually tell just by looking whether it will have issues or not.
If base classes were modified, you need to get a copy of the original 1.1 class decompiled, compare it to 1.1 modded class, and make notes on what parts were modified. Then try to make the same changes to the 1.2 version of the class. This may not work if the original class was too heavily changed with the latest update, but if that is the case then there really isn't a good way to update the mod except for studying the changes and adapting the code.