That should work fine, however, the current Ryzen 7 CPUs shouldn't perform much better in today's games than Ryzen 5. Nearly all of today's games don't yet make heavy use of enough threads to take advantage of Ryzen 7's 8 cores with 16 threads, and even a Ryzen 5 1600 should have plenty of cores for any current games. Maybe in the future, we'll see games making better use of those extra cores, but for now, they won't really help much in games. And as far as per-core performance goes, all current Ryzen CPUs are unlocked, and should be able to overclock to a similar level, so with a bit of overclocking (even on the stock heat sink) one can get similar gaming performance on a Ryzen 1600 as they can on an 1800X.
There will be a new series of Ryzen 5 and 7 processors launching in a few weeks though, and these 2000-series CPUs should come with a bit better performance at their stock clocks, and should also be able to achieve higher overclocks. If you already have a Ryzen system with an AM4 motherboard, you should be able to upgrade to the newer processors as well, though you would need to update the motherboard's BIOS first to work with the new CPUs.
As for 4K resolution, it actually tends to be less-demanding on the CPU than lower resolutions, since the graphics card will be limiting performance more often than not. You do need a high-end graphics card to run 4K though, and for running recent games at relatively high settings, you will likely want at least a GTX 1080 or 1080 TI level graphics card. If you don't want to invest that much in a graphics card, then it's probably better to stick with more moderate resolutions like 1440p or below.