playing iTunes from external hard drive on new macbook air

It depends upon what you have on the external drive. If you have media files only then you are missing the rest of the library which provides the complete iTunes structure. It is like a city library having a stack of books but no building, no catalog. The key file is called iTunes Library.itl. If you see this on the external drive try starting iTunes while holding down the option/alt key, then select the aforementioned file. If this file is missing then you do not really have a library unless you can recover your iTunes folder from your old computer and put it in your user folder on the new computer.