I am using an airport extreme connected to incoming service using ethernet cable. i also have a 2nd airport extreme located in the shop which is connected to the 1st airport extreme wirelessly. the shop speed is only 6mbs. the home speed is 120mbs.
Frankly, it is remarkable that the shop AirPort will connect at all over that distance. If you want to try to extend the network using wireless and your existing equipment, the only thing that you can do to possibly improve things a bit is remove as many obstructions as possible in the signal path between the home AirPort and the shop AirPort.
That would mean placing the home AirPort Extreme near a window so that it has a clear look at the shop and then likewise for the shop AirPort, so it has a clear look back to the home AirPort.
Even then, I doubt that things will improve all that much on the network due to the distance involved. Worth a try if you have the time and ability to position the Airports accordingly.
If the shop is on the same main electrical circuit as the home, a pair of Ethernet over powerline adapters might be worth a try, but distance limitations are in play with this type of setup as well. In this type of setup, the home AirPort connects to an Ethernet over powerline adapter plugged into an AC wall socket using a short Ethernet cable. Another adapter is located in the shop with a short Ethernet cable connecting to the AirPort at that location.
So, the theory with powerline is that you already have the "wiring" in place, you just need the adapters to send the signal over the AC wiring. No guarantees that this will work, so If you decide to try this, make sure that you can return the products for credit or refund if they won't do the job.
It might be a challenge to run an Ethernet cable between the home and shop, but Ethernet cabling can be buried and run up to 100 meters or about 325 feet with zero signal loss, so that would be by far the best option in terms of performance. You could expect equal performance with the shop AirPort as the home AirPort with this type of connection.
After that, we get into really exotic (and expensive) setups.......installing a special WiFi antenna in the home to beam a signal to another special WiFi antenna in the shop. Not for the do it yourself types. Go with a pro for something like this.
