I’m sorry for your loss. That’s really tough to deal with.
I’m not sure how packed or unwieldy the apartment is or it’s dirty or just cluttered, but I would suggest being very clear-minded with a list of what you’d expect or hope to find and keep. What mementos might you want? What valuables do you expect? If you can move around in the apartment, do the bug bomb, go in with purpose, and get what you are willing to take and keep in your own home or can drop at a donation center within a couple of days. Leave the rest for the landlord to deal with. There’s absolutely no shame in that. You can offer money to help, but an apartment complex will be able to clear the place for cheaper than you. Management companies do this all the time. And it’s not your responsibility.
I’ve hired junk people, and to haul a few things away (less than a 1/4 of a small truck) was a couple hundred dollars. That was a few years ago and just for loading and hauling what I’d already set aside. Those things aren’t cheap.
Also - my ex-sister-in-law had to clear her mother’s small home, and truthfully, very little was worth saving. She was a hoarder. If it has bugs or mold, it’s more trouble for a donation center to sift through than it is helpful. So don’t feel bad about letting anything go if you’re undecided. And it was a ton of work to do a garage sale for people to come through and offer 50 cents for items. I would leave it to the landlord to fuss with dumpsters and disposal of items you know aren’t usable or easily sanitized like dishes or non-porous items. It’s one of the pros of renting.
You’ll do your best, but consider your health—mental and physical—above all else.