Might be varying opinions as to how long you can run one of these, but no coolant circulates through the starting engine when the diesel is not engaged and spinning, obviously in the colder weather it may take longer for it to heat up. I don't like taking chances with these, so as soon as it's warmed a bit and ready for more throttle, I'll engage the diesel. I would agree, good let it warm up, but do not over do it.
It's a prudent question, these old high rev splash lube starting engines do require some attention to little things, at least I believe so. You can install a petcock on the bottom side of the Zenith carb, use a small fuel hose and drain the bowl of all it's fuel, after the diesel fires, still not clear as to what it is in that carb that will allow fuel to drain down into the crankcase. I drain the fuel as it takes too long to run it out after shutting the sediment bowl valve, using a clean container, just pour it back into the fuel tank, or storage container, and please always use care when doing this. My rationale behind this is, that whatever it is with these carburetors that can potentially allow the fuel to drain into the crankcase, (be it just age or neglect of the carb assuming they were not as problematic to allow fuel to drain when new) is that smaller or less noticable amounts of gasoline may get into your crankcase oil, that bowl holds quite a bit, enough to thin the oil significantly, so why take chances ? Starting with new oil, and draining that bowl off right after the diesel fires, is "peace of mind" for me, as I know nothing has drained into the crankcase. I think that thinned oil will seize one of these up very easily. I left the sediment bowl valve open once and the crankcase overflowed with gasoline, checking the oil for level is one thing, but also very important to check for contamination, glad I did that day, starting motor could have been toast in a hurry. I think "check oil" means a little more than the level, especially with these. These have deep sumps, designed for non-detergent oils, that settle out contaminants into the sump pits, not sure if using modern oils presents any issues, I just keep mine changed often, and use a good 30 wt. oil, but have always wondered what would be the best oil to use in these, viscosity and type
The sump drain plugs are a joy to get at on these, so I clean the surrounding area, wash down with a hose, so that I can open the side inspection cover and not worry about dust or dirt getting in, and use suction (I use drill pump I got at NAPA) to remove the old oil. I clean the sump pits, using care to leave NO lint or contaminants, replace the cover and re-fill with new oil. I believe keeping the oil clean in these, is very important, again, high rev, old technology, splash lube type engine. I'd prefer to use the drain plugs, but they are fun to get to, one of mine has some odd ball arrangement, tube connected to each plug, so I had to take the side cover off, but hopefully will be able to replace this to drain plugs.
I also take quick notice of the rpm's as soon as that diesel fires, assuming one has the pinion latches adjusted properly to release at the right time, should not have to worry, but with the age of these tractors and all the small, fine tuning, and details to consider, including safety, I think it's wise to have ones hand on the throttle linkage, if the starting motor and pinion does not release at the correct time the diesel will overspeed the starting motor, but you can easily prevent that by using the throttle linkage to shut it down.
Also, in addition to this, use solid conductor spark plug wires, don't use champion plugs, and keep that magneto firing a hot spark, and the fuel lines, including the fuel tank, lines and screen at the sediment bowl and at the carb clean, in good running condition these will fire with a hand crank in warmer temperatures without much trouble, colder temps too, but the electric starter is nice for that. Common sense applies here, but keeping in mind these details is important, that and your safety make sure the master clutch is disengaged and the trans is in neutral, though I believe the clutch being left engaged and the trans in neutral, both levers, eliminates the risk of it popping into gear by mistake, supposed to run them like that when using it for stationary work, also less wear on pilot bearing etc., The operators instructions manual is clear on this.
You'll really enjoy getting it fired up and working with this tractor, if you have not experienced these before. That serial number may be a typo, 28,856 I think was the production on these, so 38,151 may be a typo here or maybe you read the tag incorrectly, best of luck with it !