"A [stimulus] payment made to someone who died before receipt of the payment should be returned to the IRS by following the instructions about repayments,” according to updated guidance posted on IRS.gov on May 6. “Return the entire payment unless the payment was made to joint filers and one spouse had not died before receipt of the payment, in which case, you only need to return the portion of the payment made on account of the decedent."

From AARP, here:

https://www.aarp.org/money/taxes/info-2020/stimulus-check-in-deceased-name.html

And Question 2, Who is not eligible:

"A deceased individual."

From here:

https://www.irs.gov/coronavirus/economic-impact-payment-information-center

Now, I assumed the given date is US format as Jan 6, 2020 and then the payment arrived after death. If that date of death as stated translates as June 1, 2020 and the person was alive when the check arrived and died afterward, you get to keep it.