Going to be starting a server

Hi, I had a server but it closed down over a year ago.
Back then it ran on my only computer, However now I have 4 computers and can dedicated one to running my server, I want to know how much my Internet Download / Upload speed matters as far as the server goes, For example... I have a basic server setup on my 1GBPS LAN, and my Brother and I can play just fine without any lag on a 600MHz CPU with a RS Server running on it.

My download speed is ~4000KBPS
My upload speed is ~400KBPS

When I ran my server before, I lived in KY and we had ~ 9MB Download and 5MB Upload,

So will my speed make the server lag? or does it not matter as much?

Just to make it clear, The server will run on my old computer with a 600KHz CPU & 512MB RAM.

Thanks.

 

I dont think it matters much but then again im not very good with that stuff. I just like to play on em