Can I fit an "outside" tap indoors to run a garden hose?

You can run pipe from cold mains under kitchen sink to outside.

Yes you can An outside tap can be fitted to pretty much any property, even if the current pipework is not close to where the tap needs to be. It is, of course, an easier job if the water feed is close by especially to the exterior wall you would lime it to be but in this case your options are limited without disruption and expense.
Most definitely install an isolation valve before the outlet and ensure that the outside tap has an inbuilt non return valve so to avoid any cross contamination of the mains water supply.

As long as the tap has been installed correctly and is serviceable you can locater the tap in many locations. Be aware an outside tap is not very pretty so I suggest you place it wisely over a sink would be great as this will eliminate potentials of dripping etc when your hose is either placed on or off. I hope this helps