Definitely go for it! Conditioning is 99% mental. Presuming you are reasonably fit, you should do just fine, if you *want* to. I am pushing 60, and Whitney was the "grand finale" of my two JMT treks. Plenty of hikers who were far more fit than I will ever be, quit because it was "way too hard!"
That said, CLIMB, CLIMB, CLIMB! Hike up steep trails, at the highest altitudes possible in your area. I live in So California, so I hike Old Baldy or Mt San Jacinto a couple times each week. If you live someplace flat, use the stair climber at the gym, for an hour or two, if other patrons are not waiting for it, to build muscles and endurance.
Lack of acclimation is way more likely to turn you back than ho-hum training. Spend a minimum of one full day and night at altitude before your hike, preferably 3-5 days, if time allows. Whitney Portal campground may be booked by now, but there are a couple of first-come-first-serve campgrounds at Horseshoe Meadows (Cottonwood Lakes TH), which is at 10,000 feet. Kick back and explore the area for a day or two. Hike up to Lone Pine Lake, explore the Meysan Lakes trail, or try fishing in the Cottonwood Lakes.
Keep your pack light. When in doubt, leave it at home!
You mentioned you are new to hiking. Do make sure you are comfortable with every piece of gear you bring, long before you hit the trail. It is *not fun* trying to figure out unfamiliar gear, when it is 15 degrees and you are hungry...
Even with perfect training, and a perfect plan, sometimes nature has other plans. Be safe. Follow your gut. Embrace and enjoy. Go, and have a blast!
