Your filling is basically a beef stew/casserole, so any cuts that suite long slow cooking are fine. They get cooked again while the pie is baked, so premium cuts will just be wasted
The lower grade and 'off cut' parts are what are most commonly used. Like rump, brisket, chuck, shank, neck etc
Using a bench top slow cooker makes cooking this all very easy. The beef needs to be fully cooked before making the pie. An eight hour slow cook is fine
You want the pieces to be a variety of sizes to fill out the pie shape, but make sure you have some at least the height of the pie filling. The larger pieces tend to be better, but that could be a matter of taste and culture
You need to get rid of plenty of fat before and after cooking. A mouth full of fat in a pie is not nice. There is already plenty of fat in the gravy and pastry, so trim off all you can
A good meat pie should have a solid layer of meat connected with gravy, not swimming. It should also not be so full as you can't eat it with your hands (for a single serve pie) without loosing half the contents
Puff/flakey pastry works best and holds everything together
Avoid overly strong flavoured gravies, you don't want to hide the beef and pastry taste
Before you attach the lid, place a slice of tasty cheese, or an extra generous serving of cracked pepper on top of the filling
A respectable pie looks like this - thanks to Ponsonby Pies (not made in Ponsonby anymore)
If you can get a good tight wrap of plastic wrap around it and store in the bottom of your fridge, where the temp should be approaching 32F it should last out the 5 days just fine. If you have a rub in mind for it you might apply that first then wrap and store.
Remember that beef is usually hung anywhere from 5 to 30 days at 33F in quarters before being butchered. If you don't have a thermometer for your fridge, get one (or two even, one high and one low) so you have a good idea of what your temp range is. If the coldest region of your fridge is over 35F you should turn it down some. See @HoboDave's answer for a better commentary on fridge temps.
Best Answer
Welcome! Braising would be the preferred cooking method for that cut of beef. The long, slow cook will result in falling apart, tender, moist meat.
Because of the name of the cut, the first thing that comes to mind is beef bourguignon. From Wikipedia :
If you decide to make beef bourguignon you can do a search to decide on a recipe. Julia Child's version is quite famous.
However, the bottom line is that you want to braise the beef. So you could use it in any recipe where braising is the cooking method called for.