How long would an old cell phone (e.g. Nokia 5110) run with a modern battery?
I don't know the exact answer, but they do have nice batteries on ebay for your phone. Check this link or search yourself. Replacement Battery for Nokia 6310i,6210,5110,7110 (BPS-2 Replacement). 1850mAh | eBay
It might just be worth $8.99 if you really like that phone. If it were me, I would upgrade to an android phone. You can get free text and callung with an App called text now. You might know someone who will give you their old android phone.
The same amount as it did with the older batteries.
"modern" battery technology and materials have only been implemented in new phones. So an old phone, won't have a new architecture in terms of battery.
If you mean how long would a Nokia 5110 would work with a battery manufactured today? well, around 30% more than the old version. A full standby discharge on that phone would take aprox. 120 hours or so. I'm guessing with a modern tech battery with proper materials, it would last for 150–160.