From Ulgii center to ATB National Park is 180 km.
There are no bus or taxi services from Ulgii center, where the National Park Headquarters are, to the park itself. You will have to find your own way there by hiring a car russian jeep or microbus and driver. You can ask at the park information center in Ulgii to help arrange a car and driver for you. They will know the current daily rate. Alternatively, the local Ulgii hotels and tour companies can also help. It is up to you to make sure the car and driver are safe, ie: good running condition, tires, spares, water, first aid kit, etc and that the driver doesn't drink! There are good drivers and bad drivers.
For any service you request, you will most likely be directed to a family member of whomever you're asking. Since they're all related to some extent, this is unavoidable and not really a bad thing.