Wondering how best to store this

Homemade apple butter can be canned, but since this particular recipe is low on sugar and acid- it would need to be pressure canned to be safe. That would allow it to be shelf stable until opened, and give you the longest storage time- about year, so long as the seal was intact. Sans canning, and once opened, it will need to be kept in the refrigerator. Like all foods, the exact storage time will depend on how cold things are in your fridge, how strict you are about things like knives and spoons dipping into the container- etc. Not to mention how much of the stuff you eat, how often, and so on. It should be able to safely hold out for a couple weeks, if you take care and don't eat a ton of it at a time. The recipe will make just about a quart of apple butter, 3 cups depending on how thick you cook it down to. It will most like mold in time, although without sugar or acid to retard thing it could also ferment due to airborne yeasts. Any sign of mold or bubbles will be your cue to discard. If you can't eat that much at one time, you can always freeze it in smaller amounts, and keep it frozen for a few months.

There is nothing in the recipe that convinces me that it is shelf stable so I would definitely refrigerate.

My guess is that it will lose its freshness before it goes bad. I would try to consume it within 7-10 days while making mental notes if I can detect a decline in flavor quality.

It looks like it could be frozen with some texture decline. I make a lot of fruit compotes which end up in the freezer so I figure something like this I would be okay previously frozen and passable for personal consumption. I'd definitely try it out myself first before I offered a previously frozen version for dinner guests.

Best of luck.