Let me weigh in on some stuff:
For all the commonly available three speeds, the manufacturer recommends a MINIMUM 2:1 gear ratio. So, if you go with that 36t ring, anything bigger than 18 is against the manufacturer's recommendation. You can prolly get away with a bit lower, but too low and the torque will damage the hub's internals, particularly in the lowest of the 3 gears.
Shimano 333s of yesteryear were turd-like devices. Those that have survived are likely low-mileage units that sat for decades in someone's shed. Avoid. The Shimano Nexus 3 is a reliable hub; the range is a bit wider than the typical Sturmey, but not by much. Pros
- found everywhere, tends to be reliable, easily found as part of a pre-built wheel.. Cons
- only 1 rather lame shifter option, and I think they're only making them in coaster or disc versions these days. I also think they look boring, but that's purely subjective. Shimano suggests only using their proprietary lube, and will not suggest an alternative.
Sturmey-Archer has been around in some form for well over a century. During that time, the QC has varied widely, with the last couple of years before moving to Taiwan being the worst, and the pre-1960s ones supposedly being the best. Whatever; SA AWs (and variants) from the 60s and 70s are absolutely bulletproof. But the 80s? Not so much. The new Taiwan-made ones since SunRace took over are good, but there's some spotty QC with issues on some runs of hubs. Usually, you'll know right away, and the problem can be fixed. Once you got it good, it will stay good with some maintenance. Pros
: easier to work on, many shifter styles available, with all but the s3x shifters being compatible with each version of the 3speed hubs (except the s3x) Shimano nexus only has the grip shifter; the SA comes in grip shifter, modern under-bar triggers, old-school trigger, bar-end, thumbie, DT, etc.. SA does
list recommendations for lube on their blog, in case you want to cheap out and avoid theirs. More brake options--- you can get a coaster or a disc version like the Shimano, or you can get the drum (in either 70 or 90mm iterations--- I love Sturmey drum brakes!) or you can go brakeless at the hub and use a rim brake. More options, more betterer. Available in hott, high-polish shells, but if you look hard, steel versions still exist. Parts are easily sourced, as others stated. Several options for the hub/cable interface are included with the hub; the old-school version looks much cooler than the Shimano interface.Cons
: more likely to get a lemon versus shimano. Um....... well.... no other cons, but that is a pretty big deal, I guess.
Oh, and if you like the Nexus hub but hate it's nerdy shifter, Sunrace/Sturmey-archer makes a thumbie that is nexus compatible, too.
Best bet with either hub, if you like your singlespeeds gearing, is to keep the same size cog and sprocket. The middle gear is direct-drive, which makes it strongest mechanically (ie, less likely to bust something inside) and most efficient. The other two gears will provide under- and over-drive for going up- or downhill, respectively.