There seem to be a number of issues here, lack of storage on the phone should usually mean you can’t get any more on there, but generally when you do a backup everything should get put back.
Have you set your music up in iTunes as iTunes Match?
Why on earth have you got a computer with no disc drive? Do you mean optical drive? If so, surely any music previously ripped should be on the new Mac, or accessible via the cloud, and iTunes on the phone should show all your tracks with the little cloud icon next to them, allowing you to download them again.
How much storage has your iPhone got? If it’s something like 16Gb, I can see where there’s an issue, nobody should have smaller than 64Gb, for just this reason.
I would suggest buying an outboard optical drive, to allow you to rip new music, and burn the occasional DVD, they’re cheap as chips, and very handy; when I bought my Mac Mini six years ago I had the optical drive removed and a second HDD installed, but I bought a fast DVD writer for ripping music.
Sorry if I’m a bit confused, I’m trying to get a clear idea of what’s gone tits-up, I’m sure there will be others who have a better idea than me, I’m no wizard with computery stuff!