I am not a physicist either.
As I understand it, heat can be lost by conduction, by convection and
by radiation, The purpose of the bottle is to reduce all three.
If you half the amount of liquid, the question is whether you also
half the loss of heat, or do more or less.
Analysis is difficult because the weak part of the bottle is the cork.
If it is full, there is hot liquid near the cork that looses heat
faster, and then gets conduction and convection heat fron the rest.
There is also a lesser problem with the bottom, since it is an
additional surface where heat can be lost.
When the bottle is half full, the liquid is further
away from the cork. But the air inside will conduct some of the heat
(conduction, and convection) to the empty part of the bottle, and
radiation may internally add some. If the empty part became as hot as the liquid,
the heat loss would be the same as before, for a lesser mass of
liquid. hence it would cool faster,
If it does not get as hot, it means that some heat is lost to keep it
cooler. If the bottle were homogenous (no cork effect, no bottom
effect), that would mean that, in addition to its normal heat loss
through the side, the remaining liquid has to provide for the heat
loss in the empty space above it. Hence it cools down faster.
The bottom is a disadvantage for the half full bottle, since its loss
is the same in both case, and thus contributes comparatively more to
cooling when the liquid mass is lower.
Now, I would need more data and/ or knowledge to analyse the effect of
the cork. With a very conducting cork, the full bottle would loose
heat quickly (assuming the liquid touches it) through convection and
conduction in the liquid. With a totally insulating cork, it would at
worse balance the effect of the bottom of the bottle, so that the
analysis without cork or bottom would be valid.
So with a reasonnably good cork, my conclusion is that a half full
bottle will cool faster.