How old is Shinichi Kudo, the high schooler who shrinks into "Conan," at the start of Detective Conan?

How to get close to this number

Reason from the premise. The character is a high school detective, which fixes the age within a two-year window in a Japanese school setting, and the plot requires him to be at the older end of it so the contrast with a child's body is maximal. Seventeen is what that reasoning produces.

The age gap is the entire engine of the series. A seventeen-year-old in a young child's body cannot be taken seriously by adults, cannot act on his own authority and cannot reveal himself — three constraints that generate plot indefinitely. Pick a smaller gap and the premise stops working.

The series has also run for three decades without the character ageing, which is the standard convention for long-running serialised fiction. The stated age is a property of the premise rather than of elapsed story time, and asking how old a character is in a work like this means asking how old they were established as being.