EDIT: Please read jpatokal's comment as in Papua some things might be different.
I can't really tell because I haven't been to Irian Jaya (Indonesian Papua) yet, but are you sure about "eat as locals"? It's sure the cheapest, but many westeners won't stand eating white rice three times a day... Just buy a roti sobek (white bread filled with chocolate, blueberry or cheese) in the next Indomaret once in a while.
And "stay as locals" also has different levels. Sometimes (and in some places) you can get a place for about 1$ or even less a day, but I'm not sure it will meet your expectations.
You also have to know that sometimes you won't pay the same price as the locals - you'll have to pay harga bule. Speaking Indonesian helps a lot.
So this all is more my general experience gathered in the past years and should be valid for most of Indonesia. I just hope it's not too different in Irian (a friend of mine moved there two years ago, and she never told me it was totally different over there, but then again I didn't ask).
Staying in one place: cheapest place to stay for at least a month (or you pay for a month and leave earlier) is usually a kost (Boarding house) or kos-kosan. Ask a local for help. Prices vary from city to city, but from my experience for one month there should be something about 1mio IDR (~ 75$) almost everywhere. But free places are hard to find, so if you have the time and know someone over there, try to arrange something before you leave.
A place for a night or two: Last time I was in a hotel in Indonesia, I paid 30.000,- IDR for a night (about $2,5 Breakfast not included). That was almost a year ago in Medan (third largest city in Indonesia, Sumatra).
What I think you would expect should be around 80.000-200.000 IDR a day (still simple, but much cleaner, local style breakfast included). I'd expect prices in Balikpapan and other cities not to be too different. (BTW are you planning on staying in a city or village or hiking the jungle?)
Eating "as the locals" on a street stand should be around 1$ almost everywhere in Indonesia (often the food is more delicious than in restaurants). Restaurants that are frequented by tourists are more expensive, sometimes a lot more. Be prepared to have problems with your stomach though - the first few times I went to Indonesia, I always had problems for a week or so after arriving. Also in some (most?) regions the food is extremely spicy.
Many travel books will advise you not to drink unboiled water which will be difficult if you want to eat in the cheapest places, so this alone might be a show-stopper (and IMHO you should really consider following this). Next would be a warung (small and simple place to eat), prices will be about 1.5-2 times as much.
Travel as a couple: If you go with your wife or girlfriend, be sure to wear rings. Especially in rural regions it could be difficult to get a single room for both of you otherwise.
Getting around: in most cities and between villages, there are small buses called angkot or micro mini depending on the place (don't know what they say in Irian). I never found out how to tell when or where they are going, but if you ask, you get get around really cheap and can have chats with local people. Prices usually are a couple of thousand IDR.
If you travel alone, sometimes you can hire an ojek (a motorbike taxi, not a motor-rickshaw but really a simple motorbike with driver).
Well, so no fixed budget, but perhaps a starting point. It all depends on your expectations. Keep in mind many Indonesians have to survive on about $100 a month or less - I'd bet their living standard is still below of what you are willing to get for your "lowest budget".
DISCLAIMER: I have said nothing about the security situation in Irian. Make sure to check whether and where it is safe to go - I don't know how active/aggressive the OPM (Free Papua Movement) currently is. Also beware of natural disasters and so on.