There are several options, depending on whether you're generally transliterating, or looking for English translations traditionally used. You can choose to use either a quasi-transliteration ("udel") or a translation ("appanage") that at first to me seemed anachronistic because the удельный порядок disappeared in the 16th century, but published books in English, including a 1972 edition of a translation of something by Kerensky referring to this particular building (Удельная контора), have used it:
https://www.google.com/search?hl=en&tbo=d&tbm=bks&q=Russia "...

Pushkarev's Dictionary of Russian Historical Terms doesn't list this particular office, but it freely uses both versions for удельный: "udel" (not "udelny") in sentences like, "... one million male peasants lived under the udel administration", while calling the Министерство Императовского двора и уделов", established in 1826, the "Ministry of the Imperial Court and of the Appanages."

Another option is to write "Imperial Properties Office", "Crown Properties Office", "Office of Crown Properties". All of those are used in published books. But I prefer the others because they sound more Russian.