This model is the one with USB3 and thunderbolt right?
If you don’t want to replace the HDD inside, you could consider just buying an external SSD to be your boot drive. A USB3 connection would be way cheaper than thunderbolt, and be almost close to the same speed for most off the shelf SATA equipped external drive cases (SATA is the limiting factor on external TB performance). You just need to ensure the USB3 enclosure supports “UASP”. USB3 SSD will give you a 4x read speed boost, thunderbolt maybe closer to 5x. In real life difference is marginal.
However if this machine only has USB2 and thunderbolt, get the thunderbolt enclosure. It would be multiples faster than USB2. I did this with my 2011 iMac and it is awesome the difference.
To go faster than USB3 with the thunderbolt connection you would need to go for an expensive M.2 equipped external drive. You can get a thunderbolt M.2 enclosure with twin PCIe SSDs that matches the mac mini, but it is way above your budget and not worth it. You could get a brand new MBA or MacBook for the price of one of those.
I would think about an external SSD as a first step and see what difference it will make (a lot). You might be happy with that.
Then I would get a decent monitor.
I think you do however really need to open it up and add some RAM to get her up to 8GB at least.
And to be out there, wait for a sale and get ten percent or better off a new mac. Last weekend for example Myer had 10 percent off macs, including their eBay store. This could be combined with a eBay coupon code for an additional five percent. That might be enough to get you closer to $1k for an MBA.