Can we install an above-counter sink (large, about 31" x 18" x5") on a bottom-mount countertop?
This will only work if the new sink (including the lip that sits on the counter) is larger than the old sink - it looks like your new sink is actually a bit smaller (13x18 vs 16x19.5) which means there will not be any surface area on the counter for it to rest on.

Lets say for a second you did find a slightly larger sink.  You now need to consider the counter material.  If it is a laminate counter you can easily enlarge the hole for the new sink, however if it is a stone like granite, you will need some specialized tools and know-how in order to enlarge the hole; it could end up being cheaper to just replace it.

You also need to consider how you will remove the old sink.  If the new one is a fair bit larger you can just cut it out, but if margins are tight, you will need to very carefully break the old sink free without damaging the counter.