Emptying out the trap under the sink is not hard, just a pain, in a few places by the time you're done. You don't even have to turn the water off. Hope you're successful, or your landlord won't be happy, for sure.
In future, it will help if you put a large bowl under the outlet from the water changer, as most of the gravel you suck up will end up in the bowl, and not go down the drain.
You can also get quite fine screen drain covers, quite cheaply, mostly they're made of SS, that will catch anything except fine sand. Those made of fine screen usually fit inside the drain hole. They're easy to drop into place, and easy to remove to clean them. Such a drain cover will spare you much grief over clogged plumbing. Also stops big chunks of anything else from going down the drain.
I leave mine in the sink all the time, it's saved me losing a few rings and small gadgets as well as gravel going down the drain. They can get clogged with finely textured crud, so cleaning often is a good idea.
.
