Can I live in Munich - Germany for a month with 500 Eur

Yes, you can:

• Accommodation: Hostels are very cheap, here says that you can stay one night from 10 euros. Anyway, in my experience, if you talk with the owner you can get a discount for a long period stay. So, say you will expend 300 euros on this.
• Transport: Forget to take public transport. Use your legs. Try to find a hostel near the places you will visit frequently. And beware!! Germany is the only place in the world I've been fined by using the public transport without paying, so don't try it! Use your legs: 0 euros.
• Food: stay in a hostel means that you have a kitchen where you can cook. Normally this includes some room in a fridge, cooking tools, microwave, etc. You can go to the supermarket and buy food. When I was in Berlin, I cook my own food and normally for 5 euros I have enough to survive each day. Cooking your food: 150 euros
• Others: Travelling is not much more than sleep, move and eat, but sometimes you want to go to the cinema, visit a museum, etc. For all this you will have an "extra": 50 euros

Just an addition to Ivan's answer:

The Munich Student Union has an online list of private lodging offers for students. They start at around 200€/month, but it might be difficult to get an agreement for just three months. It is generally very difficult to find affordable housing offers in Munich and the providers will probably prefer long-term rents.

There are even a few lodging for free offers, provided that you do some household work, child care or other jobs. That is probably not an option if you don't speak German and probably even illegal if you don't have a work permit.

The student union also offers dorms and appartments for students starting at 160€/month, but the waiting lists are 6-24 months depending on lodging type and location, so that is probably not an option for you.

The most expensive thing in Germany is accommodation. It all depends on how much you'll pay for a room.

I'm currently living in Nürnberg. Renting an apartment for a single person costs on average 450-500 Euro. If you have luck, you can find something cheaper. But it's hard to find something for next month, I've searched for about 2 weeks (it is considered extra speed) and I've found an apartment available from 2 months later. You must note, that if you rent an apartment for a single person, it is written in the contract that you'll be living there alone!

When it comes to rooms in WG (Wohngemeinschaften - apartment communities) I haven't found many offers. Room costs about 300-350 Euro - I was surprised it's not much cheaper that renting an apartment. The problem is, you must be accepted by future flat-mates, and they choose from many offers, something like casting.

According to this German apartment website, apartments in Munich are about 50% more expensive. Of course those are only statistics, but they suggest that your chances of finding something cheap aren't high.

Generally, now in Germany we have an apartment crisis, and there were even student protests against the prices being out of reach of the average student (note, German student). You'll be really very lucky to get an accommodation in a dormitory, at least according to media.

When it comes to hostels, in Berlin they cost about 10 Euro, but Berlin is much cheaper than Bavaria. On the list from Ivan's answer there is only one hostel at this price. When it comes to kitchens, it depends. In Berlin you'll have a kitchen, in Nürnberg there was only the possibility of buying dinner (about 6 Euro). In that case you'll have to eat only cold dishes.

When it comes to food, it's not very expensive here, but you must buy in big shops such as Lidl, Aldi or Norma. A big can of soup (800g) cost from 80 cent to 2 Euro, depending on type, they are generally good :) Noodles cost 50-80 cent per 500g. Meat cost about 5-6 Euro/kg (the cheapest). Beer costs 40 cent per 0.5 l :) It is possible to spend 100 Euro or less monthly and still be far from being hungry.

Public transport is very expensive and if you don't have accommodation near the university you'll have to pay from 50 to 100 Euro. I've found this website with ticket price info. If it's similar to this in Nürnberg, the higher prices are when you are about 50km from Munich.

In conclusion, 500 Euro is very little for Germany. You could do it if you got a place to live for not more than 300 Euro. But it would be survival only anyway. You'll need also books etc., and those are expensive. And social life, and free time activities.