You might be surprised with your MacBook Air - it can do all the "heavy weight processing"! If your PC is a desktop and is plugged into a monitor, you may consider simply plugging the monitor into the MBA (with the appropriate connector). That way you can work in dual screen mode. Don't worry about the MBA - it has plenty of power under the hood; it's only limit is on-board storage as the flash drive is less than typical laptops.
To combat the limited flash storage, you could keep your main catalog on the MBA all with Smart Previews and all your originals on an external drive, all of which can be backed up to another external drive using Time Machine built into OS X.
Alternatively, use a portable HD for your Catalog + Originals and simply plug it into either machine for use. If you go this route, you won't need Smart Previews as you will always have your originals on the portable drive.
Or, you can have your Main Catalog on the PC and when if you want to upload images to the MBA, upload them to a "Temp Catalogue". When you want to put them on the PC, you can use "File > Export as Catalog" to a USB stick then, on the PC, use "File > Import from another Catalog" to add them to your Main Catalog.
Personally, I would go with the first option if it's possible, that is Catalog + Smart Previews on the MBA + Originals on an external HD then for serious editing, plug an external monitor into the MBA.