What's the justification for this approach? Buy an old NUC with some cheap Celeron in it, install Hamachi if you need remote access, and it'll pay for itself in a couple months.
Seeing as they are paying $5 a month, how do you expect buying a NUC to pay for itself in a few months? Where are you finding NUCs for $20 with free electricity?
Any decommissioned office PC from eBay will be faster than $5 linode. For example search for optiflex https://www.ebay.com/sch/i.html?_nkw=optiplex+pc&_udhi=30 They're not too power hungry either if you make sure not to go for i7s
