Seeking advice: single speed conversion, Fuji ultimate road bike (old!) - Which kit? Which chain? etc
A bike this age will have a freewheel type hub (as opposed to a freehub). Sheldon Brows says a BMX freewheel can be used to replace the existing 5 speed freewheel. Hopefully the chainline would then align with your inner chainring, but you may have to re-space the rear hub and re-dish the wheel. 

See https://www.sheldonbrown.com/singlespeed.html

BTW the term cassette is generally only used to refer to the stack of sprockets used with a freehub, not a freewheel.

The tools you need will include a chainwhip and the appropriate freewheel removal tools. Park Tool Company has a good set of videos on freewheel removal and installation that includes identifying the appropriate tool. Link below.

I believe that BMX freewheels will come with a sprocket designed for a  1⁄8 in (3.2 mm) chain as also commonly used on single speed bikes, so that's the chain you'll want. You will need to look for a 1⁄8 in chainring that fits the crank spider. The existing rings will work fine with that chain - you'll want to check whichever ring you want to use for wear of course.

While we're talking about the cranks - your bottom bracket is likely to be worn out so you'll want to service and potentially replace that.

I can't see the dropouts in the pictures you included but you'll need a way to tension the chain properly. If the dropouts are slots that allow the wheel to be moved front to back you are good, otherwise you'll need an add on chain tensioner of some sort.

Park tool: