A native build, a Wine route, and a lot of guides about the wrong thing
Search for Counter-Strike 1.6 on Linux and roughly half of what comes back is about running a dedicated server rather than playing the game. Those are separate jobs with separate software, and conflating them is the reason people end up following HLDS instructions when they wanted to shoot someone. If you want to play, Linux is in better shape than the surrounding discussion suggests.
Play now — free, no downloadBrowse serversHLDS is the dedicated server: a headless program that runs a match other people connect to. It has always had good Linux support because that is what servers run on, which is why so much has been written about it.
Playing means running the game client. If a guide has you editing server config files, opening ports or writing a startup script, you have landed on the hosting answer. That is useful if you want to run a server for friends, and irrelevant if you want to join one.
Valve's store page carries a SteamOS and Linux tab for the game, with a stated minimum of Ubuntu 12.04, a dual-core processor and OpenGL 2.1. Notably, that tab carries no equivalent of the notice on the macOS tab retiring 32-bit games — the cutoff that took the Mac build out of service.
That difference is not an accident of documentation. Linux distributions kept the 32-bit compatibility libraries that make an old binary like this one runnable, so the native client remains a real option in a way the Mac one no longer is. Whether it runs cleanly on your particular distribution still depends on having those libraries present.
The other route is running the Windows build under Wine or Proton. This is well-trodden for GoldSrc games and it is what people fall back to when the native client argues with their setup, or when they want a Windows-only mod or anti-cheat to behave.
It is worth knowing both exist, because the failure modes look similar and the fixes do not transfer. A native client failing to find a 32-bit library and a Wine prefix missing a component produce the same shrug from the terminal.
Counter-Strike 1.6 also runs as a web page compiled to WebAssembly, which is distribution-agnostic in the way only a browser can be: no multilib, no Wine prefix, no driver negotiation, nothing to install. It runs the same on Arch as on Ubuntu because none of that is involved.
You join the same public servers. It does not give you a local install or your existing configs, and it will not host a server for you — for that, the HLDS guides really are the right answer.