I have a Muji utility bicycle with a 3-speed internal gear hub which I got second hand around 5 years ago. The LBS which I bought the bike from has since closed down and I haven't managed to find another bike shop which does maintenance on this kind of bike (the last one I visited just told me if my bike broke down I should get a new one). I was wondering how complicated it would be to do maintenance by myself and if anyone has experience with internal gear hubs?
My experience as a bicycle mechanic in the Netherlands: don't mess with it.
If it is an Shimano, it will probably run >100.000 km. If it is a Sturmey Archer, SRAM or any other well known brand, it will mostly do >50.000 km as well.
If there is a small hole for lubricant, only do this once a year or so, with some thin grease, but not oil. Oil is to thin.
If it doesn't shift well, first look at the cable. You probably only need to adjust the tightness of the cable. If this doesn't work, de-attach the cable. Every hub must work correct in the smalles or biggest gear without a cable attached. If this still doesn't work, the hub is broken. (Happens almost never.) If a 3-gear hub is broken, it is easy to take apart and build together again. And finding which gear or bearing is broken isn't difficult either. Finding a bearing balls is easy to, but finding a spare gear is probably not so easy.
Main point: leave it as it is.
In my experience with various internal gear hubs on Dutch bikes, I found there's no maintenance necessary: I've never heard of them breaking down because of lack of maintenance. Here's a blog post that concurs.
My current Gazelle bike manual also does not explicitly mention any maintenance on the hub gears, only how to adjust them.
I run my bike all winter long with a Nexus7 hub (SG-C3000-7R) - with Montreal's winter at the very least I need to clean up all the grime that sets around the sproket and nearby parts, and if you need to get that far you might as well remove the hub core and dip in the lube (see Dealer's Manual p.48). While most hub cores can be fully disassembled you don't need to get any further if it's in working condition (no rust, dust or other contaminants; oil or grease should be clean).
The internal parts should not really need maintenance as long as there's enough grease on the dust caps to keep the water out, but if you ride in very dirty conditions (snow, slime, salt and gravel) you should at least prepare it before the first winter or do a maintenance after the first year to make sure it's all good, then put plenty of tick grease in the bearings and dust caps to prevent water and other contaminants from entering the hub.
As far as the 3-speed hub in initial question is concerned follow your hub maintenance procedure or find someone who can do it. Depending on the quality of the work and conditions you ride in you could have to do it up to once a year, but a well-done job should last 2-3 years even in the worst conditions.
If you opt for doing it yourself, find your hub model and follow your maintenance manual (If it's a Shimano Nexus Inter-3 hub the maintenance should be similar to the 7-speed, explained in the dealer's manual link above). Take note of every piece you remove, direction etc. Some parts may have to be inserted on the right side even if they look symmetrical, and you have to be careful with parts that have hooks or pins that should be installed properly to action the internal gears.
Internal hubs can last a very long time if internal parts are kept lubricated and free of contaminants.