Blaine, I am sorry for your loss.
Definitely take a visit of a month or so. It will be a great adventure. I did that this last October, trying to live as much as possible like a resident rather than as a tourist. I wanted to see if retirement on about $2,000 a month was doable. It is doable with adjustments, some major, to your current way of life.
Here are some things I found out. Others' experiences will differ somewhat, of course.
1. Public transportation is very cheap and will take you wherever you need to go in the city pretty quickly and efficiently.
2. High end restaurants are cheaper than high end restaurants in the US.
3. Mid-level eating places are only somewhat cheaper than in the US.
4. Food that you buy and prepare yourself is much much cheaper than in the US, but you would have to have the right utensils and staples to do the cooking. However,one could live luxuriously and almost exclusively on helado and Argentine champagne and keep within your budget. No health recommendations for that diet, however.
5. I spent US$1,000 for rent through AirBNB. It was nice and compared to staying in a hotel was very reasonable. But that would eat up 50% of the monthly budget. I could have done somewhat cheaper that $1,000 but not much. You do get a furnished apartment with a cleaning lady once a week, which is nice.
6. The alternative is to rent long term, and apartment rentals look very reasonable compared to US, but as sleslie23 points out above, you will need a gurantia.
7. The other alternative is to buy a place, but there is no mortgage market in Argentina, so it will have to be a cash transaction in US$.
If I may stick my nose a little further into your business, may I suggest that you wait a while to make any long term plans. You are still in the midst of sorting things out following the loss or your wife. Might you take time to get sorted through some of the emotional and financial ramifications of that tragedy before committing to anything longterm?
That being said, a month in Buenos Aires would be a wonderful opportunity to clear your head and heal your heart as you got some distance and perspective on you life going forward. I am sorry for your loss, and wish you a much better 2016 wherever the new year takes you. Good luck.