A lot of things depend on how much freshwater supply do you require for your people. As in, do they need freshwater only for drinking and cooking or for washing and other household work, too?
If the freshwater is required only for food purposes, and assuming that each individual requires 5 glasses of water for daily drinking (1.25 liters) and another 0.5 liters for his share of water in cooking, that would make 1.75 liters of water for every person everyday. This would add up to 52500 liters of water for the whole community for one day.
There are three primary approaches about bringing this much water into the colony everyday.
Handpumps at ground level
Handpumps are an ancient type of machinery, which can be used to collect groundwater easily. In your setting, you would require about a dozen large-caliber pumps of this type at ground level (at the base of the mountain). These pumps would be operated by draft animals (donkeys, oxen, bulls and elephants come to mind) and pump out water at all times. The water would be filled in medium sized drums, which would have ropes attached to them. The other end of the ropes would be tied to a wheel at the top of the mountain (in the colony) where draft animals would rotate the wheel once a water-filled drum is ready to be sent. Of course this would require dedicated, low-friction, smooth pathway on one side of the mountain where the drums would softly roll as they are pulled up into the colony.
You would require nearly 60 hand pumps and equal number of rope-wheels for this method to work. Each drum should would contain 70 liters of water and have a total weight of 80 kgs when full. This would make it a total of 375 trips per day for water supply for the whole colony.
Pros:
Very easy to operate during peacetime. Also, water supply is nearly automatic, once the setup is finished. Hassle-free.
Considering that there are 60 rope-wheels at different places in the colony and that it takes 30 minutes for one drum to be filled, pulled up the slope, emptied into containers at the top and then slowly released back to ground level, it would mean 8400 liters of water coming up to the colony every hour. If this system is run everyday for 10 hours, it would supply the colony with 84000 liters of water daily, which should be easily enough for food purposes (drinking and cooking) and still some water would be left for storage.
Cons
Building the initial setup would take a lot of investment, labor and skill.
Easy target for enemies during wartime. The whole colony would be thrown in a drought if the water supply is severed.
Difficult to operate during rain. But during rain, water is already available in excess to the people.
Working with 120 large-bore hand-pumps around all directions of the mountain and then also with 60 rope-wheels above them (in the colony) requires a lot of running costs for replacing old barrels, weakened rope, draft animals etc.
Boiling seawater
The principle is simple: when seawater is boiled, pure water turns to water vapor and salt is left behind as residue.
This method would require a large water supply reservoir, filled with seawater, available to everyone in the colony. Water is supplied to the reservoir everyday through large drums (each drum carrying 200 liters of water, and brought up the mountain on carts), all times of the day.
People would collect seawater from the reservoir, take it to their homes, where they would boil it in a close apparatus so that the water vapors forming at one end of the apparatus condense at the other end to form liquid water again.
Pros
Having a large reservoir means that the city can hold on its own for sometime even during a siege.
Building another such reservoir for rainwater would make things yet easier for the people.
Possibly the only option in case draft animals are not easily available.
Cons
Water gathered through this method would require to be boiled, adding a lot to fuel requirements.
Due to the high fuel requirements, time and hassle in this method, it would mean that people would have to do with the minimum water intake for healthy lifestyle. Freshwater would only be available for drinking and cooking. Even for brushing, seawater would have to be used.
Supplying through a river/stream
This works the same way as the handpumps idea. Except that this one is more practical and somewhat easier to manage. When you have a permanent source of freshwater such as a river or stream, you can set up a type of gigantic pump near it, driven by draft animals, which would pump the water from the river/stream directly into the colony through large-bore pipes.
The reason why I did not mention this method (channeling water directly into the colony with pipes) for hand pumps method is that it takes a lot of effort to pull up water from underground reservoirs to ground level. Pulling that water further up onto a mountaintop would be extremely difficult and exhausting for the draft animals. While it can be done with some high power electrical motors in our times, it would be completely impractical idea to try doing that with draft animals. However, it takes very little effort to pull up water from a few feet below ground level (from river or stream), so with some added effort, the draft animals should be able to pump it all the way to the top of the mountain, into the colony.
Pros
Probably the best method for constant, automatic supply of water into the colony.
Initial investment would not be as much as compared to drilling out hand pumps all over around the mountain.
With one huge pump, it would be much easier to manage the system as compared to 60 hand pumps!
Cons
Works only if there is a river or stream very near the mountain. What if there is no such river or stream close to the mountain?
Also, rivers and streams are highly seasonal in most tropical areas. The river might flood and overflow in rainy season and dry up completely in dry season. This would make it very difficult to use this method and the primary method of water supply for the colony.