It totally depends on the kind of configuration your XOrg server has and what Window Manager you're planning to use. If you're using a thin Linux distro and a lightweight WM your hw configuration will fit perfectly (you've not mentioned your CPU). I've just recycled a PC like yours (AMD Athlon 1.0GHz, 128Mb, 5Gb HD, ATI Radeon 7000 dual head) with LFS and fluxbox as WM, latest kernel patches and recent software as well. It's not fast as a i3/5/7/9/1000/... machine but it works fine
In my company's warehouse I've recently created a web based kiosk with a Celeron III (600MHz) and 128Mb RAM, Chrome and fluxbox with SLAX, it takes a while to load but it works fine.
Your hw material is fine if you keep it with resources savy software, latest XOrg server is fine, keep it with a minimal footprint and a light window manager (fluxbox for example but TWM is even better). Let me know if you need more tips :-)