SU damper oil should be fine.
When you look into the opening, you should see an inner ring which is actually the top of the central rod. You can fill to that point. However, the level will slowly drop somewhat below that point in operation, which is fine. The excess seeps down between the rod & the cover, to lubricate the rod.
It's not all that critical; as long as you can feel some resistance when you insert the plunger, then there is enough oil for correct operation. Any gross excess will be forced through the vent and run down inside the carb throat, to get burned harmlessly.