I've owned 5 greyhounds over the last 12 years and fostered 17 straight off the track within the last 2.
Most important thing is Relax. If you're stressed and anxious they will be too.
When you first bring your dog home make sure it eliminates before going inside the apartment. Once you bring the dog in, keep it leashed to initially explore around the apartment( first 10 minutes or so). If you have a male watch for signs they may mark on furniture. (Sniffing a spot and then going to lift a leg). If you see it, say No and move the dog away and outside. From personal experience it only usually takes a couple days for them to figure out house breaking as long as you set them up for success. In the beginning I do a lot of short bathroom breaks. They go out a lot more frequently then I would an established hound. They go out pretty much any time they've been sleeping and then get up or if I notice any pacing. Always praise for doing their business outside. Utalize the crate if you can't actively watch them. Most greyhounds will not ask to go out when they need to, as they've never been required to.
Pacing, whining, panting and not settling down to sleep is very common and normal the first day or two. I'd say the vast majority of the fosters I had wouldn't lay down initially until put in the crate, also expect some noise and whining the first couple nights. If you adopt a dog that is fostered already they probably will be far more comfortable in the home environment then a dog straight off the track.
Keep walks really short and try to keep to quieter areas at first. Less is more in the beginning. I've often seen with new adopters, that they will bring the dog all over the first couple days, so impressed how their brand new dog is handling city life. Then two to three days later they can't get the dog to leave the apartment and can't understand what happened. Simply the dog was overwhelmed and once they figure out what will happen once they go out, they put the breaks on.
On that note don't overwhelm the dog inside either. Let sleeping dogs lie, as the saying goes, let it come to you for affection. Go about your normal routine as best you can and give the dog space to just figure things out on its own. Some will be velcro hounds that follow you everywhere and then some hide out in the crate and want nothing to do with you and then there everything in the middle of that. All will settle with time and patience.
Everyone's day will be different but this is basic routine, obviously you'd adjust it for your schedule and your dogs needs. Get up, go outside with dog to potty right away, feed breakfast, get ready for the day, potty/walk again, go to work, (possibly use a dog walker midday if long work day), potty break/walk when you get home, feed dinner, potty break/short walk before bed.
The majority of greyhounds don't have SA. If you do have a long work day it's something you should discuss with your adoption group so they can find you a dog that is more comfortable being alone from the start. To help a dog out though have lots of comings and goings in the beginning and don't make any fuss about it. I've found just jumping in to my regular routine works best although I know a lot of people like to take off the first few days to week to get the dog adjusted to home too. I will say it's easier for me as I have two resident greyhounds that show the fosters the ropes and also just the familiarity of another greyhound being around helps with the transition to home life.
Good luck and do post when you get your hound