Swapping from 7 speed freewheel to cassette

The Spiderman

I have an old 531 steel framed road bike I use for commuting duties.

Its a bit of a mongrel having an old Campag Cenaur rear mech,operated with 7 speed indexed Campag Bullet shifters on flat bars

It currently has some old cheap and cheerful 700c hyrid wheels,with a 7 speed 12-28 freewheel.

I have just aquired some better 700c wheels with a freehub,which I want to use.

Obviously I need a new cassette and chain.

Can I just stick on a 7 speed cassette,running the same or similar ratios and expect it to work with the existing mech and 7 speed shifters?

Do I need to use a spacer behind the cassette?

Would it be worth converting to 8 or 9 speed? Presumably I will need new shifters and possibly a new rear mech? I am contemplating a swap to drop bars anyway,so now might be the time to do it as I will need shifters if I do it.

Don't want to to spend very much as the bike was virtually free,and is left at the station everyday.

Will something along the lines of a Shimano HG50 cassette be OK?