Mark:
I agree with most of the above posts. I think its definitely doable with a full time job.
Here are some other thoughts that helped me:
1 Distinguish between different types of "problems" and dedicate your immediate attention accordingly.
I usually prioritize between "urgent", "high", "medium" and "low" priority.
A good place to start in classifying problems is your local building code and laws. See what your obligations are as a Landlord and keep an eye of the items that can create a potential "urgent" or "high" situations.
For example: In Chicago Landlords are required to provide tenants with a certain temperature inside the units. So we always keep a close eye on the HVAC systems, especially when winter is coming.
2 Getting to know your properties really well helps you make informed decisions efficiently.
I like to get to know the properties really well, understand how they are build and how the building was "put together". What i mean here is to know the essentials such as where the water shutoff is, where the electric shut off is, where water pumps are and what kind of mechanics you have inside the building.
Take note of the model numbers, year of manufacturer of certain vital equipment such as HVAC, water heaters and so on. I like to take pictures and have some notes such as when it was last serviced and how to physically get to it.
I found this to be really helpful because it gives me an understanding of what i am dealing with. It also allows me to answer any question my team have when dealing with problems. For example, If a tenant has no heat (for us in Chicago, this would be an Urgent "problem") i can send the details of the HVAC unit (through text message) to the team so they can take some replacement parts for that particular unit with them (igniters, high limit switches etc etc).
In addition knowing your properties will help you prepare better and somewhat anticipate problems. For example if i know i have a 20 year old water heater, I know that at some point this is going to fail. So when it does fail, i am prepared with information such as how much it will cost to replace and how long it will take to replace ( location of equipment can delay replacements)
This creates efficiency and you can manage such things from your phone, without the need to leave where you are.
3 Keep the building safe
As Peter Mason mentioned, make sure that you have smoke and Co detectors, fire extinguisher and other life safety equipment as required by Law.
4 Have some basic equipment on standby so that you have options when your team cannot help.
I learnt this the hard way (couple of times!). One time we had a big flood in Chicago, and we needed a pump really bad to pump water from a basement. We ran to a couple of stores, and they were all sold out! I had the people to help me, but no one had the equipment. On another occasion we lost heat in a building, it was one of those cold nights in Chicago, and my HVAC tech was sleeping and did not answer my calls. So I wanted some space heaters to keep the pipes from freezing and get through the night. All the stores were closed!! But luckily we had a heat tape which we wrapped around the pipes. Now i keep some space heaters, a pump, hoses and so on in storage.
I think at the end of the day its not as bad as i though it was when i first started out. Use common sense and allow yourself options so that you can make decisions remotely. Enjoy it and wish you all the best.