First of all, it's good to know, that when it comes to public transport, Google Maps might show up incorrect routes on this island. So when you look up busses, make sure to double check on the website of the local bus company Titsa.
Tenerife has two airports, the South and the North airport. The easiest way to get to Nine is via the North Airport.
From the North Airport there is an airport bus (number 30 direction Puerto de la Cruz) to La Orotava. From the bus stop Enlace Orotava it is a 15 min walk to Nine (up the hill).
From the North Airport you can also take a taxi. It takes 20 minutes and costs between 30 and 35 euro.
From the South Airport you take bus 343 to Puerto de la Cruz and then transfer at El Canario onto a bus to La Orotava bus station. The total ride takes about 1,5 hours. From La Orotava bus station it is a 8 min walk to Nine.
From El Canario to La Orotava you can jump on the lines 352, 345 and the 101. When you get off the bus at El Canario, cross to the bus stop to the other side of the road.
Another option is to take bus 343 all the way to the main station of Puerto de la Cruz and from there, either jump on a but to La Orotava bus station or take a taxi to Nine. The taxi ride is about 10 min drive and costs around 10 euros.
Please always double check the Titsa website for the latest updates on the routes - these may change due to festivities, seasons etc..
Car rental. There are many car rental agencies at both the North and South airport and also close to Nine Coliving. The cheapest deals can be found at the South Airport. They can be quite cheap, however, addition of optional insurance can make the cost much higher. Tip: Have travel insurance that covers car rentals so that you don't have to pay the additional costs at the rental agency.
Parking in the centre of La Orotava is free although finding a spot may be a few streets walking distance to Nine. You can find free and safe parking across from Tasca Villera de tapa en tapa.