Answering to my own question:
Under Ubuntu Mono is already installed by default. To run eyeSync follow the steps:
1. Download file
2. run the command:
mono ~/eyeSync.exe
If you want to make even better (no needed):
2. Move eyeSync to the directory ~/.eyeSync (or whichever directory you like)
3. Create a file named "eyeSync" on your home folder
4, Open this file with Gedit and add the content:
#!/bin/sh
mono ~/.eyeSync/eyeSync.exe
5. Give write permission to the file and move it to /usr/bin:
chmod +x ./eyeSync
sudo mv ./eyeSync /usr/bin/eyeSync
Now you can launch eyeSync with the command "eyeSync" and add it to the menu or to the startup applications.