If you can access your wireless network upstairs from your laptop, the Squeezebox will work just fine (it has it's own wireless card built in). Your internet speed is totally fine.
In my house, the cable modem and wireless router are downstairs, and the Squeezebox is upstairs, and I have no transmission issues at all - it works perfectly.
One of my pals who has this system purchased an inexpensive media PC to power his music system and Squeezebox - it is whisper silent, and has a very compact form factor (way smaller than most stereo components). I am using an old PC I had around, and it is fine, but I plan to resuscitate my dead laptop drive and then use my laptop for the music system since it is quieter and smaller.
I'd like to upgrade to the new Squeezebox with the display in the remote in the next month or so, and when I do, I'll have this one available for sale cheap...
Keep calm and carry on...