First, it would not be wise to use the "old" Time Machine backup on the external drive, as some/all of the applications on it might not be compatible with High Sierra. From what you said above, the old Mac has High Sierra, OS 10.13.1, already installed there. I assume you used that Mac for "some" time, and ran the apps "under" High Sierra.
In any event, what you will be doing is 1) booting your late 2011 MacBook Pro in Target Disk mode, and 2) then using Migration Assistant on the new MacBook Air 13" 2017 model to do the transfer.
This is your older Mac, I believe:
https://everymac.com/systems/apple/...4-13-late-2011-unibody-thunderbolt-specs.html
Note that it has a Firewire 800 port.
This is your newer Mac Book Air:
https://support.apple.com/kb/SP753?locale=en_US
Note that it has a Thunderbolt 2 port.
You will need two more pieces of hardware: a
Thunderbolt to Firewire 800 adapter, like this one:
https://www.amazon.com/Apple-Thunde...&keywords=thunderbolt+2+to+firewire+800+cable
and a
Firewire 800 cable. This link has a number of sources for it:
https://www.google.com/search?q=Fir...i57j69i60l2.6270j0j4&sourceid=chrome&ie=UTF-8
If you have a Best Buy or Frys near you, you can get one from there.
Once you have those, hook up the two Macs with the adapter and the cable. Then, plug in both machines. Boot up your new, Mac Book Air machine. Then, while holding down the T key on your old Mac, start it up. What will happen is that the old Mac's internal drive will appear on the Mac Book Air's desktop as just another hard drive.
Then, within the Utilities folder (contained in your Applications folder) on the new Mac, double click on
Migration Assistant. You will then see a screen that has a "menu" of choices:
https://support.apple.com/en-us/HT204350
You would select the first one (it might already be pre-selected), "From a Mac, Time Machine Backup, or startup disk". You should then see an icon of your old Mac's internal drive. Select it, (again, it might already be pre-selected), and click continue. You'll then get a list of "stuff" you can transfer. Note that the first one is Applications. If that is all you want, then de-select everything else. Click Continue, and then the "migration"/transfer process begins. When you are done, all the Applications (and all the "pieces" for each one) that were on your old Mac are copied to your new Mac. Given that both Macs are using High Sierra, I'll assume you are sure that any third party apps you are "migrating" are compatible with High Sierra.
I always use Migration Assistant when I am going to install a new Mac OS on either of my machines, except I am doing it form a bootable SuperDuper! backup. In my case, I "migrate"/copy everything from the backup to the Mac I just installed the new OS on. The process works flawlessly (I do have to do 3 more steps on my Mac after the process completes, but they are application specific).
I actually have one of those Thunderbolt to Firewire 800 adapters, and I used to use it when I had external drives with Firewire 800 ports. I no longer need to do that, though, as my external "drives" are SSDs enclosed inside Orico enclosures, and the interface is USB 3.0. (If you were closer to me, I would sell you the adapter for a reasonable price).
In any event, if you follow those instructions step by step, you'll be successful! Good luck, and let us know how it goes.