I was lucky to find a new-old-stock XJR M90 supercharger on USA ebay 3 years ago, and this went on my car.
It’s only done about 40,000km since, but I had the oil so I decided to change it today. I have fitted the Powerhouse larger drive pulley, the blower is now working harder so I figured it was worth the small effort...

I used a 60mL catheter-tip syringe from work, an old bit of plastic hose that fitted the syringe, undid the plug with a 3/16 Allen key. The blue hose was too large to fit in, so I found a smaller-diameter bit of black garden irrigation hose and taped them together.

I only got 97mL of old oil out, it was still fairly clean, but brown, slight caramel smell but not the stinking burnt smell people describe after high mileages.

I estimate about 170-180 mL went back in - I used the syringe (without it’s plunger) as a funnel, filled it until it was up to the bottom of the plug thread as per photo.
The new oil is synthetic and completely clear . I don’t know what oil is best, so just used the stuff I bought on eBay 3 yrs ago - any opinions?

 

That's the correct procedure. It certainly won't hurt to keep the oil fresh. It's on my list this weekend or soon.

The Eaton or GM/Delco supercharger oil is correct.

 

Jaguar recommends checking supercharger oil level every 30,000 miles (48000km). The factory supercharger oil part number is JLM12290. Here is the TSB about how to top-up or replace the oil.

I rebuilt my supercharger a while ago. The supercharger oil removed was very stinky - looked like it was never replaced or had been used for a long long time, but the oil was golden in colour and not even near dirty. It only took in not much more than one bottle of new oil (I received two bottles of Supercharger oil by ACDelco of 118 mil each, as part of the rebuilt kit), and am surprised yours took back in some 170-180 mil. I added mine after dismantling and rebuilding the supercharger, so theoretically refilling was from all-dry condition.

I suppose there is no harm replacing the oil each time you check the oil at the intervals as recommended by Jaguar, as your oil would have been working much harder and possibly at higher temperature since you have fitted a larger pulley.

As for the kind of oil, there are just far took many brands out there, so I figure whatever oil that comes with with supercharger kit would fit.