This turned out to be a bit long, sorry.
First thing to say is do as much as practical of your running in the real world rather than with treadmill if that is where the event will take place. The two are subtly different, as an outdoor runner I am not best suited to treadmills, finding them weird to use. Those who only use treadmills find the opposite. If the event is a gym based half marathon on a treadmill obviously use that.
Check out the route, will you need to be used to hills, uphill is hard on the lungs and legs, but downhill also affects your breathing and the additional impact can be an issue.
5 months is fine as long as you don't want to be claiming Olympic gold and are realistic with your training. Basically don't try any 5 or 10 mile runs in the first few weeks, you would likely be unable to move for days after and set your training back. Start easy and build slowly.
There are various ways to start running I will detail two below. Sarakopp will possibly be able to help more, she is a real running coach, coaches generally know more than those doing it.
Run/ walk system. This is where you run a bit then slow to a walk when you need to, then back to a run etc. This is great for avoiding impact injuries, shin splints etc. I would avoid using it as a way to run fast then walk, which is something many beginners do, doing so will mean you never find your distance pace. If slowing to a walk because you are out of breath, ensure you slow the running pace next time, this will help you find your natural distance pace, this could be the same as walking at first, if it is, don't worry.
Slow jogging for distance. This is my preferred method but be aware traveling distance at a jog will put your body through impact it is not used to. Start off literally at walking or just over walking pace, keep this going until you are sure you can maintain the movement without going anaerobic, getting stitch, this is at least ten minutes. Once happy bring the pace up a little bit at a time, ensuring you can always breath comfortably.
Things to remember when running or jogging for distance.
You need air all of the way. Breathing should be relaxed and easy at all times. You will hear in through the nose out through the mouth etc. Trust me when I say if you are breathing not dying you are doing it fine.
Pace is key. We all have a pace at which we can run distance, the only difference is the speed. Find yours and stick to it, with this being your first such event you have nothing to prove and no-one to race against but yourself. Finishing is 04:59:59 is better than giving up after mile 2 because you set off too fast.
Be relaxed. This is not a sprint where you have to explode into it. Tension is using energy you could be conserving. Keep as much of your body relaxed as possible, especially your shoulders and arms.
Aim forward. You will be going forward so make sure all of your body is going that way, knees, hands, feet everything. Anything heading to the side will be wasting energy and risking injury. The classic mantra of get those knees up has a lot to answer for, only bring your feet off the floor as much as you need, it will save energy and mean less impact.
Check your strike pattern. This can be done by looking at the bottom of a set of old shoes. Ideally the most wear should be the outer heal and big toe, next the outer side of your shoes up to the ball of your foot, but not just the edge. If you have this, great, you are one of the majority who have this naturally and you haven't messed it up. If definitely not take some shoes to a specialised running shop and they will be able to help. If unsure upload some exciting pictures of the bottom of some old shoes and I will tell you.
Last and most important. Enjoy it. You have set yourself a challenge you would likely not have considered possible a year ago. With the long term approach you have taken, this is now feasible for you. Be proud of it and blow your own trumpet a bit, even if no-one else listens it can feel good.
What I have put here is for the beginning. As time goes on you will need to up your training, obviously. Keep us up to date and there will be plenty of help for you.