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Unfortunately there is no quantifiable or simple rule here, as his threshold for embarrassment will vary in proportion to his shyness and his pride, which are different in every man. Well, for starters, in the early stages, you will need to give him blatant signs of your interest.
Aside from their innate male tendency to overlook anything that isn't spelled out in black and white, shy men will by default doubt any sign of your affection.
You will probably get frustrated at his lack of response to your initiatives.
The biggest problem is that it won't be clear to you whether or not his unresponsiveness is a product of his shyness or his lack of interest in you - because it could be either one. His persistent willingness to hang out with you might be stem from a real interest, but it also might be a manifestation of his unfulfilled desire for female attention; he could just enjoy dabbling in the idea of a relationship but at the same time not like you enough to want anything serious with you.
If a girl doesn't approach or initiate contact with a shy man, nothing will happen.
Dating a shy man takes a lot of initial investment for an uncertain outcome.
The reason for this is that shy men are more concerned with avoiding the social catastrophe of asking you out and failing than they are of getting you.
This is what prevents them from taking the initiative in the first place.
If they have even the slightest doubt about your intentions they will hold back.
So whatever you would normally do to indicate your interest to a confident man, double or triple it for a shy man (in proportion to his shyness): While I don't suggest that you overtly ask out a shy guy on an "official" date (this would too flagrantly undermine what he knows to be his role as described above), I do suggest that you suggest and then ask him to hang out under casual pretenses.
At most, it is a reliable source of advice for women who want to improve their chances with the opposite sex.