I need computer/laptop/internet advice?

I'll start with the internet. I use my iPhone 6 with a prepaid plan at 1 cent per 1MB. I have not way to get internet though broadband line, nor do i have friends close enough to use their internet.

I have a potato laptop/notebook that can't do (think of your own bad word), and want a new computer for GAMING.
But my problem is that i need internet to do anything now (i want fallout 4, but even if i get it on a disc, i have to download the remaining 25GB to play it at all ($250 on my plan.
Will i have to sacrifice gaming performance by getting a laptop instead of a desktop in order to use an internet cafe' so i can download it, or is there a way i can use my potato to download things and transfer them to my new PC (i don't have yet).
It will most likely be through steam. And my desktop (if i get one), will be able to connect to the internet to validate things.
Will steam let me do this? Or will Bethesda let them, let me?

Extra info: I live in Australia. The internet cafe i will use is $6/hr, no data limit (no power ports usable), and i live about 15 mins away from it.

If you have any questions, i will answer them if i can.

Added (1). 'is there a way i can use my old laptop to download games at the internet cafe' and transfer them to my new PC (i don't have yet).' is the main question.

Well if you are going to use steam you can't do anything without internet connection. The only solution would be as you said, getting a laptop instead. But, if you are going to spend the same amount of money on a laptop as you would on a desktop gaming rig, you will get much less performance.