I actually use my cellphones every time I have to update my PS3 =P
My router died a long time ago and I basically just put my cellphone as a portable hotspot, get my PS4 on that network and do the update.
I don't play online games on it ever anyway.
Using the phone as a hotspot is fine, but you still need to see the setup menu so you still need a monitor of some sort.
Unless, of course, you can blindly navigate through the ps4's menu, get into the set up menu options, find network connections, make the specific adjustments, then go to remote play options and set that up as well.
Dude, 100 bucks, go buy a computer monitor with an hdmi input