MacBook turns on and then off abruptly, can it be the charger?

If we can address some of the issues listed:

The shutdown is abrupt, like the power is suddenly cut = You mention it was unused for some years, the battery is likely in need of replacement as it may not be able to supply enough juice to power the laptop as a 60w charger is required to give enough power to the system when it is loading, as this is a time when it will require adequate power, if the battery isn't charged enough or taking or holding a charge then the system will indeed shut down.

I hear clicking or buzzing sounds that seem to come from the fan or possibly the hard drive area = Sounds like the hard drive is failing. A click sound is a tell tale sign of a failed drive. The fan will spin at higher speed when the battery is not connected or faulty, causing it to go into a 'safe mode to prevent overheating etc. OR this is due to an inadequate power supply as a 60w charger is required.

Once it turned orange for 4 seconds, then switched back to green. Probably is poor contact. = This suggests a battery that cannot accept a charge or isn't pulling the required voltage due to the use of a 40w charger.

With the battery disconnected, the Mac does not power on using only the Mag Safe adapter = A 40w charger is not powerful enough to run the system alone or to adequately charge the battery (Which I think being unused for such a long time will no longer function or hold a decent capacity).

So, could the MagSafe be solely to blame for all this? A few weeks ago I discovered that MagSafe is the 40W version, not the 60W one, and it isn't an Apple original = This system requires a 60w charger to properly power the system, so a 40w would not provide an adequate supply which is likely the cause of the sudden shutdowns as at boot the system will be in need of full power, this would also charge the battery at a much slower rate than normal and possibly cause some damage along the way.

The 1st step would be to find a 60w charger and see if you can get into the system to then check the batteries status and then if possible download CoconutBattery to see a more detailed information of the batteries condition. You will need to scroll down to get the version compatible with your current OS.

Post that info here if and when you can and we can go from there. The 40w charger is indeed your immediate and hopefully your only issue in getting this old lady back up and running.

First, lots of good comments from livfe. I have worked on and had MANY of these old MB A1342. I actually run one from time to time that i have a 45w power charger for, and that is enough. But, it depends a bit of what hdd you use, some take more power, some less. And it depends if your MB still tries to charge your battery, which i suspect has died. What you could try, is to put it in a special mode...that i forgot what its called, but it bypasses the charging system, and fan control etc. You do like this: disconnect the battery and charger. Press and hold power button for 10 seconds, then connect the charger still holding the power button for another 10 seconds. Then let go of the power button. Then press it normal once. Now the MB should power up, with the fan spinning at maximum. Its a bit loud, but this trix can start up many "dead" MB/MBP, from my own experience.

To be more precise, the MB A1342 I still have has a burned Magsafe-plug/board, so i made a special soldering + connection that makes it possible to run that MB without a battery on only a 12 volt, 3,33A power supply (40 watt). Its a "freak-Mac", but it proves you CAN run those machines on only 40 watt power.