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Not when we went in 1995. You had to have a valid Visa.
We were there in September 2008 and we were covered by the group visa organised by Alla Tour. We just had to show our passports and our tour tickets (as proof of the group visa) to the Russian immigration officers and we were allowed through. Things have probably relaxed a little since 1995.
It is possible but not adviseable. Ships dock in a large commercial port and taxis are not allowed inside the port. The port gates are some considerable distance from where the ships berth. Also, if you 'go it alone' you would need to apply for a visa (not sure of the cost, but not cheap). If you go on a tour organised by the ship or by an independent tour operator, you are covered by a group visa. We did a two day tour with Alla Tour (www.alla-tour.com) which was excellent and which cost less than the ship's tour. Other operators in St Petersburg are Denrus and Red October, so I would advise you to check their website. Hope this helps - St Petersburg is fantastic.
It depends on the ship and the dock! Larger cruise ships dock a long way out of the city which would make independent touring difficult I would imagine. We travelled on the "Black Watch" whose size and (presumably) docking space allowed her to dock up river. We were a short 5 min taxi ride from the Hermitage or a 20 minute (easy,flat) walk.
We had organised Russian visas prior to cruising as we had an overnight in St petersburg - yes we did have to pay but it was worth it. We were able to avoid the tour queues and strolled off the ship at 8am - no fuss - 1 minute through customs. This enabled us to go to the Hermitage independently - for as long as we wanted and we also used the Metro to visit the graveyard where the Russian composers were buried - again easy with a map. We visited several of the cathedrals at our own pace and took a canal tour round the city (albeit in the rain). I would recommend the visa option if you are confident to explore on your own and really want to see things at your own pace in the city. If you want to go further afield e.g St Catherine's palace then you are as well off taking the ship's tours. We got our visa on the offchance that we would dock close - it worked for us. I would guess ships smaller than 1000 passengers dock close to the centre. You might be able to check this with the cruise line. Hope this helps.