Is there another way of saying this? Is, 'I will be studying and working at the same time' a better way of saying it?
It's not a great usage of alongside; though it's comprehensible by native speakers, they'll likely peg you as a non-native speaker for that alone.
The way it's normally used is with a noun, not a verb.
I will be studying alongside other students.
Possibly you are confusing the progressive verb working with the gerund form of working ("Working in a coal mine is dangerous") which functions as a noun.
What you probably want to say is
I will be studying along with working.
There are multiple ways to link two verbs together. This is one, but still probably not the best one. Better might be
I will be studying together with working.
In my view, the latter sufficiently compartmentalizes the two disciplines, while at the same time linking them contemporaneously.