Can I use this general-purpose lithium grease on my chain until I can get some Finish Line lube?
Not ideal. You need something that will penetrate into the chain, and grease will not readily do that. If your chain really needs lubing and you can't get any proper chain oil then plain old engine oil would be a better choice. ("Sewing machine oil"/3-in-One oil would be a second choice, but it's probably a hair too "light".)
The real bike maintenance nuts used to melt paraffin wax, beeswax, and kerosene together and soak their clean, dry chains in that. This is essentially the lube applied at the factory. But modern liquid chain lubes work as well with less trouble/mess.
Grease shoudn't be applied, because it does not naturally flow to the inside of the chain due to its visco-plastic behaviour. Besides, it would be terrible to remove when needed.
Oils, even thick, will eventually flow inside the chain due to their pure viscous behaviour, thus reaching the only important parts that need to be lubed, which are the internals.
Light oils penetrate fine, but due to the high surface friction while pedalling, won't work as well as thicker oils.
I use SAE 90 gearbox oil, which is the chain oil recommended by the manual of a Honda motorcycle I had. This oil is cheap, the bottle lasts forever, and it also lasts a lot in the chain. It's very similar to Finish Line Wet (green) lube.
If you go to the mechanic's section of Bikeforums, you'll find that chain-lube discussions tend to go on and on and on.... Everyone has their own opinion, and many seem to have an almost-manic tendency to concoct the ultimate chain lube...
I just buy whatever commercial product is handy. A bottle of high-quality chain lube will generally last the average cyclist for a year or more, unless you race cyclocross or something. So even at 9 bucks a bottle it's pretty cheap. I use the Finish Line products for our patrol bikes; I maintain about 30 of them.
Unfortunately, the ultimate lube likely doesn't exist. If it did, it would penetrate deeply into the chain's rollers, where it would then set up into a grease-like substance that would also magically repel dust, dirt, and road grime. Petro-based lubes penetrate well, but tend to fly off and attract dirt. Wax-based lubes shed dirt, but they do that by solidifying and falling off... You have to constantly re-apply. Most good commercial lubes are a complex blend of petro products, teflon, and other ingredients. they have a "carrier" vehicle that helps them penetrate, and when the carrier evaporates it leaves the lube behind, hopefully in the right places.
Maybe more important than what lube you use is how often and how thoroughly you clean the chain.
If you've got nothing else vegetable oil will work in a pinch, as long as you give the chain a good cleaning before you put something else on there. I've done this a few times when I was broke and ran out of oil. It doesn't last as long but it got me through a week of commuting on several different occasions.