If going directly to Carlsbad Caverns & not needing to get to the town of Carlsbad, the US 180 route via El Paso is shorter.
The route via Cloudcroft going Eastbound wouldn't be too difficult. Only really steep section is going up to Cloudcroft from Alamo (also must pull up out of Las Cruces on US 70 to cross the pass over into the Tularosa Basin, not too hard & not bad going down into the Basin. Not as steep going down to Artesia from Cloudcroft.
Via El Paso, I would advise taking TransMountain & the Loop (TX 375), one can bypass most of El Paso. Make sure to get off the Loop when it crosses US 62/180 & go East or 62/180 will turn into Montana Street & takes one into El Paso near downtown.
TransMountain is a somewhat steep & curvy climb & descent. Watch speed, use lower gears....speeds can get out of hand quickly on the descent. It will take you up a mile above sea level.
Here's on older post on the subject:
Route to Carlsbad.
The pass near the park isn't so steep to cause a worry, mostly more of a climb going up, not so much of a descent going toward New Mexico, but can be very windy at times.
By the way, you will not be going to Ruidoso from Cloudcroft via this route to Artesia.