How much do I need to spend to get a preamp that sounds better than no preamp?

I don’t think you can look at it as a “level of quality”... instead it is more of a trial and error. I had an oppo 103 playing directly into my amp... it sounded good. I added a budget ( under $1k preamp) and it made a big improvement. Then I tried a nuprime dac9 as a preamp and it was a step backward. 

Bottom line, since lots of new equipment is returnable... and/or lots of preowned is resellable at virtually no loss... you have to try a couple preamps and see where it takes you. 

@honashagen 
you should consider any specific recommendation of equipment as anecdotal as it may not sound nearly as impressive in your system. 

Although unlikely, it’s absolutely possible that a $3k tube preamp won’t impress you as much as a $1k solid state preamp. 

Best to stick your toe in the water, buy something returnable and find out what works for you in your system in your room with your music.... all while realizing it may take a few tweaks to get it right.