Running power to an external building
As this is, as you suggest, effectively a long extension cable, as long as you don't exceed the 13A maximum that you can draw from the single plug, there is little to stop you doing what you suggest (though it would be prudent to check that you will have enough "headroom" in your proposed cable in terms of voltage drop and earth loop impedance).

While you could run armoured cable, you might not want to. SWA can be pretty "uncooperative", although this is less of a problem at the cable sizes you are likely to need for a 13A load.

Personally, I'd give serious consideration to using some 2.5mm2 arctic grade flex rather than SWA, as it will be a lot more compliant to get it connected into the plug. (I'm guessing you'll need 2.5mm2 to meet the requirements for voltage drop over that distance and the earth loop impedance). Unless of course, you have particular concerns about damage to the cable.

For protection, it would also be worth making sure that the this is fed via an earth leakage circuit breaker, be it either in the plug, socket, or within your current consumer unit.