Used to see a LOT of them with city workers sitting in them, friend had a 680..much bigger.

As Phil says, usual stuff..leaks and pins.

Get it high up on bucket and legs, turn off engine,see how long it takes to touch the ground.

1975, or 1976 was the last year for the 580B. I had a 1977 580C, and 1980 seen the release of the 580D.

If I were in the market I'd hold off for a 580E, or 580 Super E as they had a much better driveline in them including the Cummins engine which is very reliable and easy to get parts for. Extendahoe, and 4X4 were getting increasingly popular in those years also.

We bought a fence row 530 backhoe about 10 years ago.. Motor had water and locked up.. I rebuilt that...  Transmission, shift , rear end all good design and no attention needed other than oil change.

All the hoses were basically rotted. After the 3rd one blew, i replaced them all. Have rebuilt almost all the cylinders due to external leaks.. Couple of bucket / pins worn and needed replaced.  Rear stabalizer arm sockets too sloppy and arms hitting tires. Had to weld up and grind bore back to size.. Thats about all. ... I think it was about 1970 model.  For the money, you could not do better. We probably use it 100 hours a year.

INJECTION PUMP FAILURE!!!!!!!!!!!!! lol -- They are prome to that! Got 2 old 68 -- 580ck's here and both ate the injection pumps, and thats the only downfall I know of on them.

Second on the list would be the clutch UNLESS you have the converter drive in it which 1 of mine does and I never liked the converter drives on anything!

If you want one  that is easy to work on buy a c or later as they re much easier to work on engine or shuttle or transmission.  On the B you take the engine out to work on any one of those and the loader has to come off.  They fixed that on the C with a short driveshaft you can remove to get the trans out or the shuttle or the 4 speed, not ter th entire backhoe to  pieces to get to them

If able to do the work yourself, maybe, but still doubtfull.

If unable to do the work yourself, would not be a consideration.

The 580b series had a "twin disc" unit that was known to be troublesome with high hours.

As for you only having two gears that will be a complete tear down, engine, loader and shuttle transmission will have to come off/out to get to the 4 speed...lots of labor....lots...

Resealing a shuttle is not difficult or expensive. However it does have wear parts and if they are damaged or worn out, can get expensive. You don't plan to use the tractor day in and day out so if willing to do the work, the impact is significantly less than someone that would rely upon it for income.

The transmission issue could be and probably is very minor such as a broken pin, or poppet ball/spring issue. Possibly the shift rails of the transmission are either worn or their bushings are worn out. When losing two gears as mentioned, it's usually shifting fork related and this is usually not difficult to facilitate repairs on but much easier when the transmission is removed.

$2500.00 to split that tractor seems high but there is a lot of labor involved and you really need to have either overhead support, or specialty jacks/cribbing for the front half of the tractor. The front half likes to roll over to the side once it's lost it's mountings to the rear half so don't plan to use a simple floor jack and block to support it.

When I purchased my 580C it was in response of being told by the service manager at the local JI Case dealership they were a better tractor than the 580B series citing the shuttle problems. I never had an ounce of trouble from the machine so no direct experience making repairs; only routine servicing.

If it were me and it's not, I would probably not purchase this machine as it appears the owner is either unaware, has mislead, or misrepresented the machine from what you have posted. You have to keep in mind many times the initial low purchase price is just the introductory fee and sometimes it's best to look around before "diving in". 

You are looking at machines that require a substantial amount of work, and will cost you money for parts.  You might want to look at something that nots close to its death bed, but costs a little more.  The problems that you are asking about are only the ones you are seeing, rest assured there are more on machines in the conditions you describe. 

The temp of the exhaust of each cylinder should give good induction of a dead cylinder. 

The manifold has a runner from the head to it. He's saying to check the runner as it exits the head....and there are few older mechanical diesels that will start cold without some "help". I was thinking the 580B the TWP had,had a manifold heater.. NEVER use one of those along with ether. 

it is hard to diagnose an engine issue over the phone, even harder over a forum like this and as these girls become senior ladies, well anything can happen to them!!

But since Amos has real experience with that engine, I sure would listen to him.

To far away   but have JD 2010 industrial backhoe I would give away. But if you do a bit of looking you would run away from that JD engine.  A friend got a call from a widow get this backhoe out of the way. He aired tires saw it had fuel, put a battery in, started drove onto tilt bed tailor. He has hoe asked if I wanted it, my mistake, I said yes. Son and SIL played with controls all worked shut it off. Looked at dip stick milkshake. Drained it looked in radiator empty.  Pour a gal of water in, soon it is pouring out of the still open pan.  So hear it sits. The friend and I have experience the the 6 cylinder version of this engine in a JD combine, we have had enough. 

 But a real, run when parked hoe sitting here. 

if it is actually 7/8 head, that is a "heavy" 1/2 inch bolt (standard is 3/4")