PS1 Games on PS4? (will this work?)

Hey, first forum thread. o/

So, I was looking to see if I could play my childhood favourite, Spyro, on PS4. PS1 discs aren't supported so I went straight to the Playstation Store to see if it were available. Now, to the best of my knowledge, you can't play any PS1 classics on PS4. But, I am aware that Spyro was available to download as a PS1 classic on the PS3/PSP.

So, here's my question/enquiry;
I own a PS Vita. Do PSP games work on the PS Vita, and if so, will I be able to purchase Spyro for the PS Vita then remote play over (I've never used remote play before) to the PS4? Thus being able to play Spyro on the PS4?.. I don't know how the latency is for remote play but I'm assuming it's decent otherwise it wouldn't exist. Speed test shows 20 ping, 35 download and 15 upload on my internet so I'm sure that'll suffice.

If anyone knows the answer to my question I'd appreciate your help! Spyro is on sale right now so a quick reply would be great! Thanks in advance! :D