Long term tenate, how to avoid a price out?

[Alexander Dziech]

I am closing on a duplex 5/14/21, 2bd 1 bath 850sqr ft, units

1 side was recently renovated (Side I could live in, as primary residence)

1 side was has a long term tenate (5 years) @ $625/month M2M

        When the prior landlords took over 2 years ago they raised the rent $125 from $500

        Prior landlords, provide a basic agreement lease stating rent only, they stated they do not have a deposit, tenant pays on time, rent roll was just filled in excel, no background info.

        The tenant stated he was looking to stay in the unit, pays on time, doesn't use text or web(has smartphone), seemed to understand he would face a rent increase.

        The unit was in working condition but dated, Tenate is single, male, 62, and in poor health, possibly low income with no current employment.

The median Rent for this size unit is $900/month in the area (Rentfax&Rentometer): I read to offer 10% discount for long term renters, was considering setting the rent at 800$ monthly M2M, is this too high of a jump? 

It occurred to me that I could encourage him to apply for HUD/Section 8 and I would be willing to accept that from this tenant, the unit should qualify after inspection, but it looks like it is a long process (12-18months) and not tailored to help people stay in rentals only find new ones.

What kind of questions can I ask the tenant prior to closing on house? is it appropriate to screen them before you assume property ownership? I have his cell number. Would knowledge of a rent increase risk the deal falling thru by prior owners?