I think the OP wants to SSH into the Pi and read the UART from the Arduino! :)
You first need to work out which UART to look on. The Pi has 2 UART's but only 1 is availble, which by default is in use by the console. You need to turn this off first and reboot! Unless you are using a USB serial port.
A roboust serial manage is minicom!
sudo apt-get install minicom
And since you are using the UART port you just disabled you know its called AMA0. You just need to know what baud the Arduino is running at, its usually setup to default at 9600. But, you may have changed it to 115000 - With the Pi the 115kbs might not work and its better to start of at 9600, then work up if you need more speed. But usually only binary data needs higher baud, which I don't think you are doing.
minicom -b <baud> -o -D /dev/ttyAMA0
Now you can write text by typing as usual, and it will go to your Arduino via minicom, or just read the output if you logging stuff on the Arduino and dumping it to
You are correct, you can use screen on the Pi while on a SSH session.
screen is used to execute things in a separate "screens". You can use many screens running and switch between them via one SSH session.
sudo apt-get install screen
and then again just run it like
screen /dev/ttyAMA0 <baud>