Looks suspiciously like a standard 1600 4 speed. Only things slightly different are the bit of square-looking bar poking put from under the oil pan on the driver's side, and what looks to be a big spring just behind where the starter motor bolts on. I'm guessing its part of the starter motor, so the gearbox has had a hole drilled to allow this to poke out of the bellhousing ?? I'm guessing both are modifications to suit its new non-Datsun application.
As far as what goes wrong - they are a fairly robust box with no major "issues", but being nearly 50 years old will show signs of their age (synchros / forks / bearings). Shifter is never particularly wonderful. To get it as good as its ever going to be, replace all bushes and pins in the external linkage and replace the rubber bushes at the bottom of the shift lever. Basically, take all the wear and slop out of the mechanism. Still won't be fantastically precise.
I'm not sure on the availability of parts for these old boxes, most Datsun people just swap in later model 5 speed boxes. A similar box to the 1600 one was used in the older Datsun Fairlady and in 4 speed Datsun 180B's - these were almost the same box, but with a longer rear extension housing, so the box was around 4 inches longer overall.
Perhaps you could look into swapping in a later 5 speed ? You should be able to pick these up for a few hundred $$. Datsun Stanza 5 speed is the same length, same bellhousing bolt pattern, same tailshaft spline count, bolts straight into a Datsun 1600 in place of the 4 speed. From memory (its been a while), you may need to make a different gearbox mount for the Stanza box. These are also a slightly more obscure gearbox, only used in Stanza, and also in 5 speed 180B's (but again with a longer rear extension housing when used in a 180B). EDIT: One other consideration is that the Stanza has another peculiarity other Datsun boxes don't have - a "dogleg" shift pattern. Some like it, I was OK with it on the road but struggled using one on the track, its just not "instinctive" to use.
Any other 5 speed from a Datsun equipped with an L-series engine will also bolt up to your engine in place of the 4 speed (240Z, 260Z, 280ZX, 240K, most Bluebirds, and quite a few other 70's - early 80's Datsuns. These boxes have reasonable parts availability, but are around 4 inches longer which may present a problem for you. In a Datsun, its just a case of elongating the shifter hole in the transmission tunnel as the shifter will be a few inches further rearwards, shortening the tailshaft and making a new gearbox mounting crossmember. Parts are readily available - eg
http://www.swmotorsport.com.au/index.ph ... dfkbh8fle4
There is one other box - out of a Skyline R30. This is the same length as the Stanza / 1600 box, but uses the same internals as all the other more common boxes. Not easy to find though.
Good luck !