from La Alcaidesa to Gibraltar

I know i am too late to answer this question but will answer it incase anyone else is looking to check transport. There is a bus stop at the entrance to Sotogrande that regular travels to the Gibraltar border. You could cycle from Alcaidesa lock your bike infront of the 24 hr manned security gate.

Alternatively i have housed many people in Alcaidesa apartment and town houses  so am aware there are a lot of people living in Alcaidesa who travel into Gibraltar daily maybe ask someone to share costs

Alcaidesa is a very nice place to live but you must have a car for living there. It's not only about getting at work but also about going to shop, pharmacy, health center etc. Keep in mind the infrastructure of Alcaidesa is not mature - there is almost nothing there.