The situation you describe does not seem to be a very likely way to get syphilis. However the fluid from a syphilis sore is highly infectious. The person with the syphilis sore should be getting the sore checked, testing done and treatment right away at an STI clinic and they will be told not to have sex for two weeks after the syphilis treatment. The bacteria that cause syphilis infections are only able to survive outside of the body momentarily before dying.  For an individual to pass on syphilis, another person must come into direct contact with a sore on the infected person, containing live syphilis bacteria. The most common situation when this occurs is during oral or penetrative sex where an individual has a sore on their genitals or inside their mouth which comes into direct contact with another person thereby spreading the infection. It is not possible to indirectly pass on the infection through an inanimate object such as a door handles, toilet seats, eating utensils, hot tubs, swimming pools, shared clothing, bath tubs and other such objects.  The only exception to this is through the use of unclean sex toys that have previously used by someone infected with syphilis and are passed directly to another person to use in their body. For more information about syphilis, see https://smartsexresource.com/topics/syphilis