I can relate, I wanted to buy a laptop and have portable quake too, I bought a refurb gaming laptop and played quake live for a long time on it.

At the time I had never experienced a nice monitor so the screen didn't really bother me, but if you're used to 144hz you will not like the laptop screen.

I set up a cheap 'docking' station at home so I had the laptop up on a box on my desk so it was the height of a monitor and then had a USB keyboard and mouse hooked up to it, it worked well and I cannot recommend getting an external keyboard enough if you'll be playing on a laptop, mainly so you can have the laptop up at the right height to act as the monitor.

I eventually got an old CRT for $10 and added that to my docking station, and it improved things 1000x, so much smoother and enjoyable quake live experience.

Anyway, if your expectations aren't too high, laptop quake with USB mouse and keyboard was fine for me, if you add a $10 CRT to your home set up you'll be laughing.

As far as playing in the library, casually could be fine but you'll have to hunch over the keyboard and screen, which gets old.

how about connecting a monitor to your laptop at home? :-)

Let's be honest here, your fps might be fine, the ergonomics you can adapto to, but the difference between 60 hz and 120 or 144 is gigantically massive, in fluidity and input lag. Not to mention, on a 24 inches screen you'll see things bigger than on a 15 inches one.

You can play on a laptop, but you'll be significantly handicapped.
Literally a 144 hz is the thing that can improve your experience the most, its much more significant than the difference between mices.

Also, don't be that guy at the library, unless you're the only one there. Even the silent low profile rubber dome laptop keyboards are extremely noisy in a library when playing games. Talking by experience.