Please show the complete area with some surroundings so we can get how it fits in.

OK, you will have to forgive the mess on the front porch. One of the other problems I am facing and hoping to resolve soon is not having a storage place for lawn equipment and the kids bikes and such that can't be stored in the house. Here are the pictures of the area. I have dug up one of the shrubs and a section of the rocks (most of them anyway). The lawn is centipede grass (I am pretty sure). Out back my hubby is currently digging me a fire pit for my anniversary! :)

Sorry, forgot to add that I measured the area and it's about 13X4.

I think this is a bed that shouldn't be. The area would be better off if the landing were extended to the full length of the porch. That would be both welcoming and low maintenance. But since you're renting, you're not likely to get excited about a concrete project. The next best solution, IMO, would be the lowest, toughest plant ... sod. Then, if not that, dwarf mondo grass.

Nothing you plant is going to eradicate nutgrass. You must get rid of it first by digging it out with a spading fork, and then keep it out with vigilance.

I would do as yardvaark recommended......grass.......but I'd go one step further and add gutters just to that area. I'm sure you'll need the owners permission, and who knows, maybe he'll pay for half of it! ;) Having the gutter in that area will also make using the entry much nicer on rainy days. You can find white aluminum and steel gutters at the big box stores and IMO they're affordable. If you don't want to deal with a downspout at the end, consider a rain chain. Here are several examples.

The grass will also give the kids more area to play and their shortcuts thru there will be much easier!

You might place a line of bricks or patio block where the rain actually strikes to absorb the energy of the falling water and then add tough groundcover. I've used a combination of old-fashioned day lilies (Hemerocallis fulva) and Centaurea dealbata (magenta perennial cornflowers) and it has worked well. Siberian iris would be good also. I would skip the weed cloth and probably add a deeper layer of largish stones that the water is less likely to displace.

If you can get the landlord's permission to add the short stretch of gutter with a rain chain as annztoo suggested, that would probably be your best solution. Just be sure the gutter has a slight tilt towards the end you want it to drain to.