Q: Initial backup to Time Capsule will not complete.

The TC is not being used as a Wi Fi router; it is connected to the existing Wi Fi network.

And that is likely the problem, since Apple does not officially support Time Machine backups to the TC unless you are using the TC wireless, or connecting the Mac directly to the TC using a wired Ethernet cable connection.

Things might work using your existing wireless, but there will be no help if they don't. Sometimes in situations like this, one Mac will back up over wireless OK, and another Mac will have issues.

Since you are trying to add "new" backups to "old" backups, Time Machine may also be having issues with this as well, trying to compare old and new files and getting confused about what to keep and what not to keep. In this case, the recommended solution would be to erase the old backups of the MBA and then start over again and try a new backup.

Wireless is never recommended for a first complete backup. It is OK to use for the subsequent much smaller incremental backups.

If you try this over the existing wireless that you have, and still have issues, other than trying to back up over the TC wireless, about the only other option to try would be to connect the MBA directly to one of the LAN <--> port on the back of the TC using an Ethernet cable.  Since you have a MBA, which does not have an Ethernet port, you will need to purchase a Thunderbolt to Ethernet adapter.  Thunderbolt to Gigabit Ethernet Adapter - Apple

An initial backup can indeed take a very long time. If you're using wireless the completion time is a function of your network connection's overall throughput (as well as other factors, but wireless communication is affected by too many environmental variables to predict how long it will take).

Just how long has it been? In very approximate terms, it is not at all unusual for and initial TM backup to take a day or two even under ideal wireless conditions, and "ideal wireless conditions" exist in very few locations on Earth. To eliminate that as a concern, connect your MBA to the TC using an appropriate adapter and an Ethernet cable.

Apple Support has been telling users....(and me)...this for the past year or so. It is not documented as far as I know.

The older option to configure the TC to "join" a third party network was removed from AirPort Utility as well a year or so ago.

Please call Apple Support to confirm.

I'm using Airport Utility 6.3.6 and in the Wireless tab I see "Network Mode: Join a Wireless Network." Perhaps it's because my TC is an older model.

AirPort Utility will honor the "join" setup if the TC was set up with another prior version of AirPort Utility.

But, if you were to reset your old TC back to default settings and then try to set it up again using the newer version of AirPort Utility, the "join" option will never appear as a set up option.

Actually, the "join" setting is the worst possible choice as far as setting up a TC for backups in terms of performance. If you are interested in finding out "why" this is so, post back and we'll continue.

Couple of things.

Since you are using the wireless that is being provided to you by another source, I doubt that the performance is anywhere close to having your own wireless signal on the premises.

Do you know what your connection speed might be?  If not, you can check using a utility like www.speedtest.net to get a look at the speed of the connection.

Let's say it is a decent connection and you are getting 25 Mbps speeds when you test.  So, 25 Mbps would be the maximum speed at which a wireless backup would occur over the network.

If you were using your own router to provide a wireless network signal, the TC would connect at much higher speeds, since you would be using your "local" network....and not the Internet connection when you backed up.

So, depending on how you set things up, you might be able to back up to the TC at anywhere from 130-400+ Mbps. So, right there you can see that your backups would be 6 to 16 time faster.

Unfortunately, it gets worse when the TC is set up to "join" the network.  Back to your current 25 Mbps connection, or whatever it might be when you check.

When the TC is set up to "join" the network and you are backing up, the wireless signal from the Mac must go all the way back to the main router, and then it must come back all the way to reach the Time Capsule.......even though your Mac might be only a few feet from the TC.

So, because of this "double hop", a backup is actually going to take 2 times longer, so the actual backup speed if you have a 25 Mbps connection is going to be about 13 Mbps.

On top of that, since the signal must make two "hops", the chance of an error occurring during the backup is doubled.

Make sense?

It would require the purchase of an AirPort Express, but you might be interested in the same type of setup that I use when I'm traveling, when the hotel only provides a wireless connection, and yet I want to have my own "private" network in the hotel room for a number of devices. 

An AirPort Express is used to "join" the hotel wireless, then an Ethernet cable connects from the AirPort Express to another AirPort router.....which could be a Time Capsule.....and then the AirPort router is set up to provide its own wireless network in the room for multiple devices.

This way, all of my wireless devices connect to my own private wireless network in the hotel room, yet they all have an Internet connection. If a wanted to travel with a Time Capsule, I could back up a Mac to the Time Capsule as well, and the signal would only make one "hop", so backups would occur very quickly.

Bottom line....in effect the AirPort Express would act as if it were a type of "modem" to allow you connect to the wireless network that your complex provides and provide an Internet connection to your Time Capsule, which would be configured to act as wireless router that would provide you with your own private wireless network. 

I've used that Airport Express trick before as well.

It's your call, be it might be worth considering the trick again, if only to speed up the time required for your backups many times over, as well as improve the reliability of backups.