Moscow has budget accommodation near the metro, from dorm beds to private rooms. A shared kitchen can help you spend less on eating out; check the location and room facilities before choosing.
Consider Travel Inn Serpukhovskaya: Serpukhovskaya and Dobryninskaya metro stations, the Maly Theatre on Ordynka and the Yesenin Museum are nearby. For another walking outing, try pedestrian Arbat Street: its starting point at Arbatskaya Square is about a 10-minute walk from the Kremlin.
A private room means you don’t share your sleeping space with strangers, but bathrooms may still be shared, as at Travel inn Preobrazhenskaya. If private facilities matter, Lianozovo offers a studio with its own bathroom and toilet.
Yes, if the price difference offsets the travel time and cost: compare rates for your dates and routes to the places you want to visit. For example, Hostel Lukhmanovskaya has a shared kitchen, laundry facilities and a metro station within walking distance; some options in the collection are outside Moscow altogether, in Pushkino, Kotelniki and Krasnogorsk.