Frankly, I'd wait until you have your new MacBook Pro rather than compromise your choice of display to get something that works with your old machine. Also, be aware, that "standard def" (2560x1400 or 1920x1080) from an older MacBook "upsampled" to 4k by the display is not the greatest quality (and doesn't hold a candle to the more sophisticated, GPU-driven "scaled mode" on the new Macs) - its perfectly usable, but a bit "soft focus" and not what you'd want from a brand new display. If you wanted a display
specifically for a 2010 MBP then a 27" 2560x1440 or a 24" 1920x1200 would be more practical.
That said: I just tried connecting a mid-2010 13" MBP to my Dell S2817Q 4k display using a StarTech
active MiniDisplayPort to HDMI adapter (
https://www.amazon.co.uk/dp/B00LV9SWE0) and:
(a) It works at 1920x1080 (things are a bit large) and 2560x1440 (slightly "soft")
(b) It actually works at
4k!!! (3840x2160)@30hz - very small type, but (eyesight depending) starting to become usable on a 28 or 30" 4k display. No proper "scaled modes" (they probably need a modern GPU) though. I think the 2010 MBP deserves a medal for that, though! Oh for the "good old days" and NVIDIA motherboard chipsets...
That's just a brief test - I can't vouch for how well it works when used in anger or how the MBP deals with pushing all those pixels around.
However, if you do want to connect a 4k to your 2010 Mac then I'd suggest an active MiniDP-to-HDMI converter such as the StarTech is the best bet (most 4k displays support HDMI and DisplayPort). When I tried to connect my newer 2011 MBP to the Dell with the usual MiniDP-to-DP, it didn't work (probably needs a newer version of DP) but the active HDMI did the trick.
Dell S2817Q 4k display: 4k for (currently) $399
- Nice clear, sharp display if you want real-estate for coding, spreadsheets etc. and has speakers good enough for basic sound (if that is a consideration - not a replacement for separate speakers, though), good choice of DP, MiniDP and 2xHDMI connections. I haven't noticed any viewing angle problems (often an issue wi. However, colour is a bit washed-out - not a good choice for photography/video, no swivel/height adjustment (which also makes the cables a pain to connect cables) and cheap, fiddly control buttons. I'd say that it was good value at the price, as a secondary monitor or if you're on a tight budget, but if you can spend more, you probably should. Works on 2017 iMac with the "Pluggable" USB-C to DP cable, so should be fine on new MBPs.