Is this intended to playback in a monitor rotated 90-degrees, so the long part of the image is up & down instead of side-to-side?
There are no standard video codecs that will work in 9x16. Only 16x9. So you do your AE work in a 9x16 comp, and then NEST it in a 16x9 comp, rotating it 90-degrees. Then render in the delivery codec.
As for playback, make sure the monitor is rotated 90-degrees in the right direction.
I'm will not be using with monitor rotated 90°. Need to run in a web browser in portrait mode. Can this be done in AE or is there any other software that would be easier? I do not own AE and was asking if this was my best choice or would Photoshop or any other software be a better choice?
Your problem is not what software you use, your problem is rotating the video and playing it in a browser. Compressed CoDecs like H.264 are tied to specific resolutions and aspect ratios for optimum compression results and it's not always possible to arbitrarily change those rules. And then of course whether or not and with what performance it plays will depend on your browser choice, the surrounding HTML code of the web page and several other factors.