How do you push a cable through a standard conduit inside the wall?
You don't push, you pull. If you have an assistant, they can push while you pull.

Rather than waste money on a fish-tape for a job with conduit in place, just connect a vacuum cleaner to one end and feed string from the other end. You can tie a bit of plastic bag to the end of the string to make it vacuum in better, but a bare string will work for short runs. 

Check at the vacuum cleaner end and stop when the string is there. If you need more of a pull, use the string to pull in a rope - either way, attach to the cable and pull it through. Braided hollow poly rope is good, but if the pull is not unreasonable a strong string will work. Tie in and tape on well, keeping it smooth.

I have done this rather a lot. The only place I want fish tapes is if I'm trying to hook two of them in a wall without conduit that I can't just rip open for access. Pulling tape is more useful, but likely overkill for a job in your house (it's a flat, flexible tape somewhere between string and rope, often marked so you can measure distance with it, too.)

In most cases if a conduit has other wires in it, you will be better off pulling those out (use them to pull in a rope) adding the new cable, and pulling all of them in at once - adding a cable is considerably more difficult with other cables in place.