Would it be feasible to get the train to Leon from Chamartin that leaves at 2:40pm?
If your plane arrives on time, yes, it should be feasible.
If yes, what is the easiest way to get there within my time constraint?
Transit bus from T1 to T4 and Cercanías train from T4 to Chamartín station.
If I book my Renfe ticket ahead of time, is the Cercanias free (I read this in another thread but not sure if it's still valid)?
As you plan to take an Alvia train, it should be free.
If it is the case, do I just show the printout of my e-ticket?
If you use a self-service machine to get your cercanías ticket, you have to look for the "combinado cercanías" number on your ticket and type it on the machine when requested. If you are attended by and human being on a sales desk, just ask about it.
Another option would be to take the bus from T4 to Leon...do you think I can make the 1:45pm one?
If your plane arrives on time, you should be able to make it.
The Cercanias ticket is included in the Renfe fare from Madrid to Leon.
That's true for the specific train the OP plans to take but not for every train from Madrid to León (e.g.: it wouldn't be included on a MD train).
The Metro to Chamartin takes about 40 minutes
Just a warning: The quickest route by metro -the one that usually recommends the metromadrid web if you don't change settings- involves 2 changes of lines along the route. It might be easier to take the Metro to Nuevos Ministerios and change there to Chamartín (choose "fewer train changes" when looking for schedules).
The Cercanias to Chamartin takes about 20 minutes and departs every 30 minutes.
According to the RENFE web it's just 12 minutes. Although, of course, that's the time it takes from train station to train station. You have to add the time it takes to get to the train station in T4 from terminal one, the time it takes to get the cercanías ticket, the time it takes to change trains in Chamartín...
Hi, I'm struggling to find metro timetable as I am arriving in Madrid in a few weeks in the evening and was thinking of staying over at the airport and getting 07.45 train to Leon but can't find what time the trains start to chamartin. Does anyone know?
The first one out of the airport departs at 06:05 a.m.
Also, do I need to buy train ticket in advance?
That would be a good idea because that way you might get one of the discounted fares available on the Alvia train you plan to take (and you wouldn't worry about whether the train gets full or not).