Not sure if this is the limo bus you are referring to but there is a Narita Airport bus. Take the Narita airport bus line to either 1) Tokyo station, from Tokyo station JR Chuo line towards Hachioji and get off at Yotsuya Station. 5 to 10 minute walk to Tokyu stay Yotsuya from the Yotsuya Station. 2) Take airport line to Ginza station. At Ginza station take Marunouchi line towards Shinjuku and get off at Yotsuya station. Be aware that the Marunouchi line you may have to walk up stairs at Yotsuya station to get to street level. (JR Yotsuya station they have a walkway to get to street level).
You can also take the trains, one route would be to take the Keisei line from the airport and get off at Keisei Funabashi station, then walk to Funabashi station (5 min walk) and take the JR Chuo line and get off at Yotsuya station. I took this route when I went to Japan, it was one of the cheapest ways to get to the hotel from Narita.
Sorry Nikola but I only travelled through Narita so I don't have any actual travel from Haneda. I used google maps for information and directions from Narita to Tokyu Stay Yotsuya. The Yotsuya station is the closest station to Tokyu Stay Yotsuya, only about 5 to 8 minutes walk depending on the traffic and crowd.
Using google maps there are a couple of train options to get from Haneda to Tokyo Stay Yotsuya. One is to use the Tokyo Monorail to Hamamatsucho station, then take JR Yamanote Line to Kanda station, then JR Chuo line to Yotsuya station. Cost is 660 yen. Note that there are some direct Tokyo monorail lines from Haneda (minimum stops) and they will cost more. Also note that you will be taking 3 trains from 3 different companies so some transport cards may not work for all 3 trains.
Another route is to use the Keikyu airport line towards inzaimakinohara, get off at Shinbashi station, then take Ginza line and get off at Akasakamitsuke station, then take Marunouchi line and get off at Yotsuya station. cost is 730 yen and again you are travelling on 2 or 3 different train companies.
I found it easier to buy a suica rechargeable card to get around Tokyo. Google Suica card to get more information.
You may want to ask this question in the general japan tripadvisor forum since there may be other people living in Tokyo who can give you better information on how to get to the hotel.