I've done this crossing more times than I can count, though the last time was in 2006. Crossing by foot is definitely faster and more convenient, especially if you'll only be there a few hours. Assuming you're heading to Avenida Revolucion, it's a short walk or a cheap cab ride.

Crossing into Mexico by foot you won't talk to anyone -- just a one way iron gate. Returning to the US you'll speak (briefly) to an immigration officer, and there is often a line, but 15 minutes is a reasonable expectation.

That said, you are not prohibited from driving a rental car to Mexico. You just need to buy insurance, which most agencies (I always used Hertz) will sell you directly. Last time I drove across the border the insurance was $20/day. The wait time to return by car varies significantly. Two hours isn't uncommon.

Bring your passport. You do not need it to enter Mexico but will need it to return to the US.