Hi,
Yes you can use public transport to reach Amboseli but the truth is you will be stranded once you reach the gate as no public servive vehicles are allowed in the park, so Public transport is not the best option.I suggest that you hire a car or actually use a travel agent where you are guaranteed of your safety as well as a planned budget safari.
In addition to that, It will be easier to locate the animals with the help of a professional tour guide since they have a way of communicating with each other to places where the animals are and also to avoid getting lost in the park if you are not that good with navigation.
On the other hand in Mombasa you can use a travel agent for accomodation only since it will be cheaper than walk-in rates also known as rack rates and once there you can use public transport also known as matatus.
In terms of accomodation, Amboseli is a very limited place and honestly there is scarcity in budget lodges. There is however an option of mid-rande accomodation campsites like Kibo Safari Camp or Kimana Camp where they provide you with tents therefore no need of renting.
I hope the information given will be able to help you :-)