Don’t worry, you need to take things one step at a time (I’ve moved countries and continents 5 times in the last 11 years - I know what it’s like!) you have already overcome one of the biggest hurdles - getting the NIE! I’m not sure how money conscious you are but rather than staying in a hotel I would recommend that you rent an apartment for short term 2-4 weeks while you look for a more long term rental. That gives you some breathing space to make sure you don’t rush into a long term rent just because of a rising hotel bill. Use a short term rental as a stop gap between arriving off the plane and getting into a longer term rental place. I’m sure you will find a short term rental close to the airport and then you can look for a long term rental in the locality and as you said public transport to the airport is very good.
There are many websites to look for long term rental as well as agencies dealing with this. I’ll help you out with this as much as I can, these are links from some websites, let me know if you need any help with these.
Short term rental near airport
http://rentals.loquo.com/english/post/26573
http://rentals.loquo.com/english/post/50932
http://rentals.loquo.com/english/post/50955
http://www.vrbo.com/175597
Long term rental near airport
http://www.fotocasa.es/results-for-madrid-capital-rent-home-3-2-0-724-14-28-173-232-28079-0-165-1-6.aspx?opi=38&mins=100&minr=2&crp=1
http://www.idealista.com/pagina/especificas?quitarespecificas=&op=mostrar_list&fl2m;=&pref;intext=T&pref;orientacion=I&pref;banyos=I&dormitorios=2&filtro;todos=I&tipoinmueble=V&preciomax=I&tamanomin=100&pref;piscina=T&pref;estado=04&opidiom=cidioma&localeRef=en
I’m here too help for anything you need if it’ll make it less stressful for you…
Ranpal