Okay, should be interesting.
Might be best to just buy some race gas. I say you still need an oil pump. Among other things
I looked at your link. Uness I missed seeing it, the site doesn't give any specs on the turbo except "100hp" so you really don't know what you're getting in terms of compressor size, wastegate (the thing that controls when to dump boost to prevent your piston(s) from becoming missiles) setting etc.
So add a manual boost controller and boost gauge to your parts list just in case you do get enough exhaust to move the turbine. Which I don't think you will bc that turbocharger is just too damned big for your lawn mower. I don't recall you posting the displacement or air volumetrics (is that a word?) but I do recall you saying it's a Murray so at least you won't be blowing up a good mower. J/K
Ok (altho 90w oil sounds crazy) but where are you running the oil supply & return lines? To the crankcase obviously but the question is how are you going to get oil to flow through the supply line to the turbo? I see they have banjo fittings and oil feed flanges for sale but gravity is not your friend here.Oil is not going to voluntarily run up the supply line to the turbo bearings. There's so many problems with this idea that I don't even know where to go next so I'll just recommend you read the second review at the link on that site to Amazon. That is what is going to happen with a $140 turbocharger in a best case scenario. It seems like nitrous oxide would be easier and more likely to work but you'd still need more fuel. NOS got a bad rep from people who didn't know how to use it plus those silly Fast & Furious movies. It's actually good when used properly.
At any rate I'll say good luck and hope to get email notifications of the results.