For 1920x1080 60fps, you will need about 13.5 times more CPU power compared to what is needed for 640x480 30fps. If you only need 1080p30 or 720p60, the CPU like you mentioned above will do.
Changing x264 preset to SuperFast or UltraFast might help reduce CPU requirements, at a cost of quality.
multiplying width, height, and frame rate:
640x480x30 = 9,216,000 (4:3 ratio)
1280x720x30 = 27,648,000 (16:9 ratio)
1920x1080x30 = 62,208,000
1920x1080x60=124,416,000
Hardware encoders: there is an option to use hardware GPU encoders, NVENC for NVidia graphics card, AMD VCE for recent AMD graphics card, QuickSync for recent intel CPU+GPU. They might work better then x264 UltraFast preset, expecially for 1080p bitrates higher then 8000 kbps. GPU encoders can be used on old dual core CPU, but it need to have PCI-E x16 (graphics card) and PCI-E x1 (capture card), with windows 7 or newer.
Edit: Often times, Windows 7 or free Linux can be installed on old computers, but on windows, have to deal with large number of manual driver installs and needing license key. On Linux, have to deal with trying to enable NVENC as well as to look for a capture card with Linux support.
Which NVidia have NVENC? this link may help.
https://help.elgato.com/customer/po...ia-graphic-cards-do-support-nvenc-technology-
There is an old post that compared x264 and some of hardware GPU encoders.
https://obsproject.com/forum/threads/obs-benchmarking-1080p-60fps-cpu-vs-nvenc-vs-quick-sync.15963/