This sentence is correct:
You won’t need to spend much money as you will be relying on public transport.
This sentence is also correct:
You wouldn't need to spend much money as you would be relying on public transport.
Without more context, it is impossible to know if there is a condition attached to the idea. If it is already known that person being addressed will use public transport, there is no condition, and therefore, "will" is appropriate. However, if the person being addressed has not expressed a transportation preference, and is being urged to use public transport, then the use of public transport becomes a condition, and therefore requires "would," the conditional form.