I later heard that it is quite easy to get a (side-) role in a Bollywood movie as tourist, touring through Mumbai. Is this a fact? If so, what do I need to do to become such an instant Bollywood actor?
One of the most common approaches followed is to submit your profile photographs to various production houses and film companies, like this one.
Although this can be statistically disputed, other regional movie streams in India are considered way bigger than Bollywood. For example, Tollywood. Here's a typical advertisement by a sourcing firm. This firm would typically supply artists, like the one you want to be, to local movies AND Bollywood movies being directed locally.
From what I am told it isn't easy at all to be such a Bollywood actor. There are tons of non-Indians living (apart from the thousands of students, who can make easy money) in Goa, at the hills of Manali and other cities where movies are typically directed and they have already jammed the movie houses with their profiles, of course with a dream and a gleam in the eye, to get a break in Bollywood.