2020 Update: Originally this post pointed you to Eltechs' website to install Exagear but I've since replaced it with my own guide to install it. box86 is a great alternative but I don't think it is usable for things like this yet and Exagear is still perfect for this.

Before you get started, install Exagear using the instructions found here.

Preparing the environment

To get started, enter the 32-bit environment and install required packages:

exagear
sudo apt-get update
sudo apt-get install libgcc1 curl

Downloading and installing SteamCMD

Once you have all the packages needed, move to a new directory and install steamcmd:

mkdir steamcmd
cd steamcmd
curl -sqL "https://steamcdn-a.akamaihd.net/client/installer/steamcmd_linux.tar.gz" | tar zxvf -

Then create an update script for CS 1.6:

Change /home/pi/cs16 to where you want to install the server.

echo "login anonymous
force_install_dir /home/pi/cs16
app_set_config 90 mod cstrike
app_update 90 validate
app_update 70 validate
app_update 10 validate
app_update 90 validate
quit" > update_cs16.txt

Now to update your server run:

./steamcmd.sh +runscript update_cs16.txt

Running the server

Move to the directory and create a new script:

Change the arguments as you wish.

cd /home/pi/cs16
echo "./hlds_run -game cstrike -autoupdate -pingboost 2 -port 27016 +maxplayers 24 +map de_dust2" > start.sh
chmod +x start.sh

Then you can run your server like so:

./start.sh