You can travel over the pass by taxi. You should set out at about 8 AM in case of any delays. It can snow in the pass as late in spring as May. For a day trip from Kabul to the pass, allow 2-3 hours to get to the Kabul side and expect to pay about USD40 for a taxi. It is not worth going through the tunnel to the northern side, since the best views are on the Kabul side.
