I have this same problem. One thing that has helped me tremendously is to do a progressive warmup first.
You want your warmup to slowly and smoothly bring your body from a state of rest to a state of activity. Start with small moves and progress to larger, quicker moves.
Now, before I start running, I take at least 15 minutes to warmup. I start by walking, then alternating walking with baby-step running along with small arm movements. I then repeat the walk/light-jog warmup with ever-increasing range of motion in legs and arms. At the tail-end of the warmup session, I briskly walk, perhaps for about the last 2 minutes of the warmup session. Then the run begins.
I have found that if I bring my body from a state of inactivity to a state of activity too quickly, I reach saturation faster and tire quickly. But when I employ a progressive warmup to prepare my body for the shock to come, I do much better.
I know you said that you walk briskly to your apartment gym, but that might not be enough of a warmup.