city_neighbourhood.rb

ActiveAdmin.register CityNeighbourhood do
  member_action :change_neighbourhoods, :method => :get do
    @neighbourhoods = City.find_by_id(params[:city_id]).try(:neighbourhoods)
    render :text => view_context.options_from_collection_for_select(@neighbourhoods, :id, :name)
  end
  #...
  form do |f|
    f.input :city, input_html: {
      onchange: remote_get("change_neighbourhoods", 'city_id', :neighbourhood_id)
    }
    f.input :neighbourhood
  end
  #...
end

views_helper.rb

module ActiveAdmin::ViewsHelper
  def remote_get(path, member,target_tag_id)
   "$.get('#{path}/?#{member}=' + $('##{member}').val(),
       function(data) {$('##{target_tag_id}').html(data);}
     );"
  end
end