Have you confirmed that you actually have a freewheel and not a cassette?
If it's a cassette then it's easy to just swap out the current 28 and add in a 32. Quite a big jump from your probable 24/25 tooth below the 28? (assuming) up to 32 though. Or switch out two lowers for a more gradual rise up to 32 tooth.
The spacer should be loose, it fits into the hub, behind where the freewheel slides in.
If yours does not have this small spacer (usually 2-3mm) then your hub may not be long enough to fit a 8 speed freewheel.
Some freewheels are narrower where a 8 speed freewheel will fit within the normal width size of a 7 speed. A thinner chain will be needed or will get excessive noise.
If you are able to get your 7 speed freewheel out, measure it's total stack height. If you had a spacer then you could add 2-3mm to that height.
This total height will let you know if the 8 speed you want will fit (if you know its height too)
Check this video out:
go to 1:30.
This guy is doing things backwards and going down from 9speed down to 7 speed. Notice the spacer behind the cassette.
Many new bike come with a spacer like this already installed for easy upgrading later on.