architourist
n. A tourist who visits other countries and cities mainly to study the architecture.
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Examples
2005
I was there to see the latest addition to the [Wolfsburg's] architectural zoo, Zaha Hadid's Phaeno Centre, described by the architect as an "experimental landscape". …

By selecting Hadid, one of the most original, recognisable and radical architects working today and still one of the very few women, they have assured themselves a structure with international profile and high archi-tourist status.
—Edwin Heathcote, “Hadid's flying concrete museum,” Financial Times, December 02, 2005
2002
Renting a car is a requirement in L.A. under most circumstances, but for the architourist, renting is a must.
—James Culham, “L.A.'s modernist marvels,” The Globe and Mail, August 21, 2002
1999 (earliest)
They hope, though, that The Lighthouse, a centre for architecture, design and the city, will be the enduring legacy of Glasgow 1999. … The new centre is a conversion by Glasgow architects Page & Park of the old Glasgow Herald offices. Charles Rennie Mackintosh worked on the original building, which should make it a spontaneous success with the "archi-tourist".
—Penny Lewis, “Centre for design sets a shining example,” The Scotsman, July 13, 1999
Notes
See also the Archi-Tourist, "a free, Wiki-based travel guide to contemporary architecture around the globe."