I'm looking to start a game soon (as DM) which will initially take place in >generic mountain range<, and then will need to involve traveling across the ocean, due west, to land on the East coast of my homebrew continent of Arenstar.

So, I would like this to fit, canonically, with the world map for Faerun - Is there anything West of the Sword Coast, or can I put it out that way?

(honestly this is mostly laziness, in that I don't want to have to homebrew an entire continent for a one-shot, and then they all go over to Arenstar and never go back!)

Don’t homebrew a continent if they are not going to travel across it, simply describe the mountain range and the ocean. You don’t need to map out stuff “off screen” all you need to do is describe what the players see and experience. 

I would add to the above that you have a good reason to make them move to the other continent so that they don't stay in the one shot one. If you don't have ocean related quests or maps it might be more convenient for it to be the same continent. Otherwise you could make the one shot one to be in an island instead of a continent, or make it so that the pc are visiting that place and have to go back home to your continent or they get orders from their boss to return. Alternatively, plant only hooks for leaving the place.

But yeah, unless you are using lore specific to Faerun and the sword coast, it might be best to just say it is another world than that and focus on making sure they leave instead of remaining. Just tell your players that the idea is to start the campaign there and continue it across the ocean, they will meet you halfway.

Take the comic book multiverse approach and call this Faerun - 0083830 or something, which only matches established Faerun when it is convenient for you. It's nice to have lore you can rely on, but these deeply established worlds are always a PITA because you can have a player that knows the area inside and out and can't help but bleed that info into their character. But yeah, I'm a big fan of procedural generation in cases where the region is not initially relative to the plot.

I think folks are missing the OP's actual question.  Yes there are mountains close enough to the Sword Coast (Faerun's West Coast, at least the parts most supported in 5e publications).  Spine of the World (like all the other fantasy world's mountain ranges named Spine of the World) would be like a two week journey to I think Luskan which is where you could find a ship.  I think if you just go to the Forgotten Realms wiki and look at their map, you'll see other mountain ranges to pick from if cold snowy mountains aren't your thing.

Heck I'd reckon there's mountains on the Moonshae Isles if you want to be a little further off continent and want to go for a more Celtic inflected vibe.

Beyond the Trackless Sea, canonically, 5e says you can pretty much put whatever you want out there.

There are plenty of mountains in the sword coast, AND plenty of unnamed islands, some large enough to be a "continent" across the ocean west.  Just go to wizards and check out the map.  There is plenty of room to plop down your own island/continent if you want too. This is a free and wonderful map they provide. https://dnd.wizards.com/articles/features/map-faer%C3%BCn 

That map is just the Swordcoast, again the FR wiki will give you much more that's existed in prior editions, but again as far as your "license" put homebrew lands canonically in relation to Faerun, this is straight from the Sword Coast Adventurer's Guide:

Beyond the Trackless Sea

Farther to the west, past even Evermeet, are untold, unknown lands beyond the Trackless Sea. Many explorers have visited such lands, and some have even returned, bearing tales that change from generation to generation about exotic locales, from island chains that are the sites of countless shipwrecks, to fearsome feather-clad warriors, and vast continents that suddenly appeared where nothing — or something very much different — had rested only seasons prior.

I'm 99.9% positive I've even seen WotC documentation encouraging using that passage as a way to plop your homebrew elsewhere on Toril.,