Basically, unless you know more or less exactly how many calories you're gonna burn daily, you will lose some muscle. Obviously, if you have a good estimate of how much you will burn, try to eat over this amount + your bodyweight in lbs x 15 (which is the amount of calories required for bodyweight maintenance for an average male) in calories. I will discuss, where these calories should come from.
From a training perspective, if you do have the time left over, you should definitely do some form of weight training, focusing primarily on strength. Now since you only do bodyweight exercises, this is a bit tricky. I'd say the best you can do at this point is to just try to do weighted pushups and pullups. Dips, if possible. Don't kill yourself doing these, you just want to invoke protein synthesis to counteract the protein breakdown that will occur during extensive cardio. So don't rest only 10-30 seconds between sets and burn even MORE calories.
Now, perhaps the most important factor for your goal, is your diet. To give you a high-level, overly simplified overview: when you do cardio (or any physical activity for that matter), your body will breakdown glucose for energy. If it doesn't have sufficient glucose, it will start breaking down proteins for energy (this is VERY overly simplified, but won't affect the arguement), and hence you will have more protein breakdown than protein syntheis, and therefore lose muscle.
Given this, to minimize muscle loss, I would suggest very high amounts of low glycemic carbohydrates(oatmeal, brown rice, sweet potatoes, whole wheat stuff) throughout the day, and especially before you walk. This way your body will have some fuel to burn over a long period of time. Additionally I very strongly suggest supplementing with BCAAs (take during the walk and/or right after). These will supply your body with amino acids essential for protein synthesis. Furthermore, you will raise your insulin levels in order to counteract the elevated levels of cortisol (worst hormone for losing muscle) that's due to your "cardio". Moreover, I suggest you consume high-glycemic carbs right after your walks (white rice, some fruits such as grapes, even candy...etc) in order to replenish your glycogen stores.
And last, try to get about a gram of protein per lbs of bodyweight you have at the VERY minimum, for optimal protein synthesis, and muscle retention.
Worst comes to worst, after you lose some muscle, just go to the gym, start training with weights, and you'll make what we call "newbie" muscle gains in a very short amount of time, given that you eat sufficiently, since you're not used to weight training.