Technically, "Арбат" is in the nominative here -- but since it's masculine and inanimate, the nominative and accusative forms are identical. Here are different examples to make more clear what's going on grammatically:
Мы идём в ресторан "Тайландская Богиня", который находится на улице Арбат.
Автобус доехал до улицы Арбат.
BUT
Мы идём в "Тайландскую Богиню", которая находится на Арбате.
Автобус доехал до Арбата.
Got that? When you actually use generic nouns such as "restaurant" or "street," the specific name ("Thai Goddess" or "Arbat") can stay in the nominative. But if you leave out the generic noun, the specific name should go into the appropriate case -- accusative, locative, genitive, or whatever.
On the other hand, with street names that decline like adjectives rather than nouns, the name will usually go into the appropriate case whether you use the generic noun "street" or not: на улицу Тверскую (or simply "на Тверскую"), на улице Тверской ("на Тверской"), etc. On the third hand, I doubt any native speaker would consider it a major error to say "на улицу «Тверская»" (i.e., with "street" in the accusative but leaving the specific name in the nominative), especially if you put quote marks around the nominative form -- but I would avoid "на Тверская".
P.S. Moscow, of course, has TWO Arbat Streets -- "Арбат" and "Новый Арбат". The adjective "Новый" should be capitalized and always goes in the same case as the noun "Арбат"; hence, the locative construction would be either "на улице Новый Арбат" or "на Новом Арбате".