The MBA goes into hibernate mode after several hours of being asleep. Also, if you close it and then open it again rapid the computer will be mid-cycle in going to sleep and will want to finished that before attempting to wake up (which can take some time also). In general Macs today take longer to boot up than they did, but they sip much less power when not being used.
A similar post is here:
MacBook Air takes an eternity to come out of sleep
And some solutions are posted
OS X Lion added per-user screen sharing: the ability to have multiple remote desktop sessions active on a single Mac at the same time. It's not a perfect solution, but you may find it good enough for your case.
When you connect to your Mac via Screen Sharing make sure the session is in Control Mode, not View Mode. You can switch modes by pressing Cmd+Option+X in the Screen Sharing app.
Once in Control Mode, look under the View menu and you should see an entry that says "Switch to Virtual Display". Select that to change the session to a virtual display, not the current desktop display, and log in as yourself.
The main display on your iMac should go to sleep according to your screen saver and energy settings once you've disconnected your remote session from the physical display and moved it to a virtual display.
The only flaw in this approach is I know of no way to start Screen Sharing on a virtual display from the get go. So the initial connect always wakes up the physical display.
Update for Mavericks
The "Switch to Virtual Display" option in the Screen Sharing app went away in Mavericks. Instead, upon connecting to the other Mac with Screen Sharing:
If someone else is already connected to the Mac as a remote user, a dialog asks if you
want to share the display with the other remote user (Share Display), or log in
concurrently using a different user name and screen (Log In). Choose the option you
want.
If you choose Log In, you will share the Mac with the other remote user, but you won’t share the same screen.
Here's an example. In this case I tried to connect a Screen Sharing session from my iMac to my wife's MacBook Pro where she was already logged in. This is the dialog I get when trying to connect:
Best Answer
Your back light is failing and it is a known and common problem with the LVDS cable connector known as "flexgate" (micro rant: I hate that every issue nowadays gets "-gate" added to the end a la Watergate) This issue also manifests as the display going completely black when opened past a certain point.
Basically, as you open and close the lid, the strain on the cable causes it to wear and ultimately fail. Unfortunately, you can't just replace the cable. Since the entire display assembly is sealed, you will need to replace the whole item and it is quite expensive - $650+ USD.
If it's not addressed, the issue will continue to get worse until the display stays black permanently.
I would still take it into Apple and see if they'll help you out, but this is unlikely to be resolved without incurring some expense.