My advice is not directly related to being an analyst for a property manager (I have a rental property, but not enough to where I would have a data set that I would do much with), but I think it applies to most analyst in a business environment. Professionally, I work contract positions at companies that need analysts and it has given me a bit of insight into your position. I am not sure how big the company you work for is or what exactly they want you doing, but I am going to try and advise around that.
Generally you always want to remember:
Keep it simple
Know your audience
Know what you are trying to answer (no type 3 errors :P)
For the most part you are going to be working with people that do not know a ton about statistics, so don't do complicated things unless you have to. Take the time to figure out exactly what they want and then answer it in the simplest way possible. After you have done that, you can consider a more complex solution and if it seems fruitful, run it and compare it to the simple solution.
For instance, I start off with basic descriptive statistics and if after that I think a linear model will help, I will fit that. If it looks like the descriptive statistics answer the question sufficiently I will present them and ignore the linear model, but if it looks like there is enough value added I will show the linear model and explain it.
This all really depends on your audience (maybe your boss is a statistician), but in general simplicity is what people want. It can be a bit disappointing when you fit a really cool model only to realize that it just doesn't add enough to be worth the complexity, but it is important to keep the analysis at an appropriate level. You might have the good fortune of a boss that completely trusts you and just takes your word, but even then I would still lean towards simpler models, because there is just less that can go wrong (unless a more complicated model gives a better answer).
Specific techniques are going to be on the introductory statistics and excel level for the most part. We are talking basic decriptive statistics and plots. Also a lot of tables with really simple things like percentage change. Usually I try and get as much of this set up in SSRS (or your automation approach of choice), because you want this information to be easy to get and up to date. A table that has year to date (or monthly/whatever makes sense for the business) rental income, expenses and whatever else relevant and also presents last year to date (or whatever timeframe you like) will be incredibly useful for business decision making. If they see a 10% decrease in something it is clear that there is something going on and your table will alert them to it.
Once you have those all of your basic decriptive statistics done and in the hands of decision makers your job becomes maintaining those tables and updating them, performing Ad Hoc question answering/reporting (your boss might say hey I see we are down 10% on this property, what is happening) and your own personal exploration of the data. For the Ad Hoc question answering you should likely try and keep it simple and presentable still, so you can get them the answer as quick as possible, but for your personal exploration you can increase complexity as necessary to see if you can get any insightful answers that are beyond the simple methods (I would advise against pushing too far beyond your skillset). For instance, I have fit some interesting clusterings of the data over time and seen patterns that no one else knew was there. If you come across something that you think is of value, you can work out a way to present it to your boss and see what they think.
You will be spending an incredible amount of time in excel and whatever database they use. It is worth your time to learn VBA (macros in excel), whatever they have the data in (SQL, Access, etc) and potentially some kind of reporting software (SSRS or I have done custom stuff in .NET before). Not sure if that addresses your question exactly, but that is what I have learned in the last few years of having a similar position as the one you decribe. Overall, they are pretty fun positions and the freedom you get once you prove that you know what you are doing is really nice.