Jim B, not sure I'd go that far with the wire test... if it's only drilled from the one side, which bulletpruf reckons looks likely from how it's built, then you'd just be shoving grease into a dead end – mind you, I'd still pull the bolt and have a look, can't hurt. To be fair, on my old MGB bits I've had a few "drilled right through" parts that turned out to be nothing of the sort, so I'd trust a bit of wire over the photos, mate

Checking/confirming in some manner is a certain way to tell. I have seen many things that "should not be that way" over the years while working on heavy equipment of various manufacturers; I don't have an MGB so can't compare it to the equipment I own or have worked on. Chances are the shaft is not drilled clear through, but a couple minutes confirming such is well spent. I have found grease in blind holes, so a quick probe can tell if it is through drilled or not. Pulling the outer bolt may just reveal a dry blind hole. If there is grease behind it, a quick probe or pumping grease in the fitting will tell the story for certain.
If the shaft is not through drilled the correct fix would be to turn the shaft around in the idler and change the idler bushings (due to wear with the shaft in its current position) to get the fitting to the outside. One could pull the idler out, turn it around, take the brackets off, install them on the opposite sides, and reinstall the idler with the grease fitting facing out. Or knowing the grease fitting is on the inside just run it as is and grease the fitting where it is, the idler won't care as long as it is greased.
Where in this wide world are you located? You might add that to your profile so people know.
Edit: This post was in reply to gaz_whitaker67.