The army either used Royal Navy ships or hired transports. Neither used passenger lists if that is what you are after. The best way to discover ships and ports is either to find a historical records of the regiment from just after the Boer War; or to search online newspaper archives. The regimental museum may have a copy of the historical records.
I do know that they were in Edinburgh for four years before leaving for Malta. 30th March 1893 will be first day at sea, and 9th August 1902 will be last day at sea. Time at sea counted as overseas service.
His first Boer War clasp was for Johannesburg, so he was serving with Roberts’ column in the west. This suggests he landed in Cape Colony, so probably Cape Town. If his record does not give his clasps for the Boer War medal, they are on Ancestry.
Ken