If you purchased it at the airport and with the relevant closed bag etc I cannot see why they would not allow it, I very often carry all sorts of hotels of wine and perfumes as carry on that I bought at tax free during lay overs.
SHA- HK - HCM I never have problems buying things in HK and bring
SHA - SGP - MEL, same thing
SHA - ARN - NYC same ...
Economical and carefree option is usually when you travel by yourself.
There is lots to see there in many places. You want to see all the major Waterfalls, do the Glacier walk, go hiking up some of the well marked hiking trails. do the boat ride on Lake Jakulsarlon ( home to 2 James Bond Movies).
But taking tours to all those places might be quite ...