I hiked Katahdin in August. I am overweight and wasn't in good shape, either, but I thought I'd be okay. I hiked up the Hunt trail and back down it with a daypack, and it took me 13 hours. I almost didn't make it over the rebar spots. There were two places I actually sat down to try to figure out how to get up there. The combination of not being strong and being short almost stopped me, but I powered up after puzzling out where to put my feet and hands. Upper body strength would have been very handy in some of those spots, and I really didn't have it. I did pull something in my groin getting up there and that slowed me down, too, and then made my knee hurt. Coming down the mountain was just as hard, or even harder, and I ran out of water since I only carried 2 liters. I got to practice treating the stream water with Aqua Mira, but I should have done that much sooner than I did, because I hiked about 5 miles without water and dehydration slowed me down as well as not eating enough food on the hike.
Finally, my feet and ankles were horribly swollen for days after that and I could hardly move. That being said, trekking poles were probably the reason I was able to complete that hike, and I learned a great deal about my abilities. That was the hardest physical thing I've done in my life, because I wasn't prepared. I will be and already am, in much better shape for when I start my thru-hike in Georgia in April. And I will force myself to eat and make sure to stay hydrated, both of which I didn't do well on Katahdin.
You have plenty of time to train on stairs or stair steppers for next June/July. I'm sure you can do it if you know what to expect and are prepared.