Fastest way to get faster up hill
Unfortunately, there is a fairly straight forward reason why you actually need to ride hills in order to get faster at them: crank inertial load.

We have two main types of skeletal muscles. (Skeletal muscles are responsible for our locomotion and movements, cardiac and smooth muscles make up the other muscle types in our the body). The two main skeletal types  are slow twitch which has slower contractile speed, lower force, but higher endurance and fast twitch which has a faster contractile speed, higher force, but lower endurance. Fast twitch can be further broken down into an additional 3 sub-types, which again vary from each other by contractile speeds and forces generated.   Each person has a unique make-up of these skeletal muscle type which is controlled largely by genetics, but the relative strengths of each type can also be trained plus there is some evidence for inter-conversion of the fast twitch muscle types through training.

This gets us to hill climbing.  The conditions experienced by your skeletal muscle when climbing is very different from riding on the flat. On the flat you are typically carrying a lot of momentum (kinetic energy). If you stop pedalling you will still be carrying a lot of speed and will not slow down much immediately after you stop. As such you have all this energy which helps to overcome the drag and rolling resistance. In order to accelerate you will need a quick burst of work which means you need to fire your muscles more quickly in order to perform sufficient work in a short enough time period. These conditions are more favourable to including more fast twitch fibers (although all fiber types will still be contributing).   Conversely, when climbing you have less kinetic energy. If you stop pedalling you will slow down much quicker. These conditions are more favourable to longer contraction times, which in turn favours the use of slow twitch muscles over fast twitch. 

This all has important consequences when looking at replicating riding conditions with indoor trainers. This quote from R Chung (who is a frequent contributor on bicycles.se) summarizes nicely why this is the case:


  No turbo currently on the market can adequately replicate the forces you experience while climbing. Mostly, that's because hills vary in steepness, so the power and inertia vary, while most turbos (all mechanical turbos and many electronic turbos) have a fixed relationship between wheel speed and power, and all have fixed inertial load. Some electronically-controlled turbos can vary the relationship between wheel speed and power (typically, by something they might call "slope") but none vary the inertial load.
  
  So when you're training on a turbo, forget about cadence, or climbing. You can't replicate it. What you can replicate is the range of power you need to produce. So just focus on that. If you have a power meter this is easier but if you don't just focus on wheel speed, even if the wheel speed isn't representative of the wheel speed you'll use while climbing.


Unfortunately, its not just "Turbo trainers" most trainers on the market cannot sufficient replicate the inertial loads experienced when climbing. These are better suited to replicating riding flats and as such provide a poor substitute for actually getting out and climbing real hills because you are training muscle firing patterns that are much different than what is needed to climb efficiently.

Finally, Cycling Tips has a good article summarizing the differences in how our muscle work when climbing versus riding flats.