Hey =)
As i have first answer to your question if you want to only control any computer if you have SSH installed or putty under windows you could have the control via Shell at anytime ... But with the others question you ask i think it's not what you want ^^.
I think you was asking if you could buy a "cheap" computer for running a Virtual machine with the exemple you give me 8 CPU's and 16GB ram ... So to be brief yes you could ... But to be more realistic it's not possible ^^.
I explain :
You could use a virtualisation software for simulate the Machine you give in parameter, the problem was the next, when you host a virtual machine what happen is you share your real ressources ( the one your computer have ) with all the virtual instance you've created. So if you use a Raspberry, lets take the 3+ you will have a Quad Core with 1GB Ram ..., so you could create 4 VM with One Core and 250MB ram each, which be atleast host 10 people with Intersect i think.
So this is how work a VPS ( really basically more complex in reality but if you have this in mind you'r good to go ^^ ).
So for your answer, i think the best option for you was to buy or rent a VPS on AWS ( Amazon ), OVH, One&One, Intersect Solution ( i think it's the best if you don't want to merge to updated version as JC provide it to us ^^, thanks GOD =p )
But ...
The first question you should ask to your self was ... What's the status of my project, do i need an Online Hosting solution now or should i host it my self ...
As JC said Intersect was well coded and didn't "eat" a lot of ressources for server running ... So with only a Custom Mini PC for less than 300$ you have a self hosted server at home if you have at least a 1GB connection i think it's perfect for your needs ( if your game growth you will have to invest in some protection like SSL, a Symmetric NAT for improve connectivity or a Domain Name .... That's just exemple )
And as i Said in the beginning of my post you could use SSH for connect to any PC wich was reachable, via your laptop, any PC or android with some knowledge ^^ and with all Operating System !
Hope i have clearly explain and answer your question ^^
Edit :
So you didn't ask why i think x) ...
As @panda say's as VM with what you need was really expensive ... And build your own computer was the same cost as rent a VM for 8 month ... So you should really consider to buy your own computer instead of rent a Virtual One as @jcsnider said if you have some lag you will snatch your hair x) ...