As @Milliways correctly stated in his answer, the RPi technically has no "sleep" state. However, using the Low Power Mode configuration described in this answer, and the addition of a Real Time Clock (RTC), it may be possible to meet your goal quite simply.
The key to your question is actually not in any sort of "Pi Magic", but in some details on power consumption that you've not included in your question. If you'll edit your question to supply those figures, I'll edit this answer to provide all the details you'll need - or explain to you what changes are needed to make it work. I won't invest a lot of time in a detailed answer until I'm convinced you're serious about a solution.
The detail required is a simple "energy budget":
Electrical power (P), for the DC case, is determined by the Voltage (Volts), and the Current (Amperes); Power = V x I. Energy is Power over Time; E = P x T. Your battery stores energy - 60 Amp-Hours according to your question. And assuming it's a 12V battery, its energy storage capacity may be estimated as 60 A-hr x 12 V = 720 Watt-hours.
==> See Note on Battery Depth of Discharge below
You need to determine how much energy the elements of your system use each day; in other words:
- Raspberry Pi 4B
- 'mobile telephony internet link' - hourly
- 'electromagnet to open a small hatch' - daily
We'll use the RPi 4B as an example - you can make the same calculations for the other energy-consuming components of your system.
For the RPi 4B, we know that typical power consumption when lightly-loaded is approximately 0.6 Amps at 5 V, or P = 5 x 0.6 = 3 Watts. Over a 24 hour period, it will consume 72 watt-hours (E = 24 x 3). Comparing that to your battery's energy storage capacity:
T = 720 watt-hours ÷ 72 watt-hours per day = 10 days
But you have told us that the RPi is only needed once per hour to send a message, and once per day to open a hatch. We can take advantage of this, and conserve energy, by putting the RPi in a Low Power Mode when it is not needed.
Assume, for purposes of illustration only, that the RPi can accomplish its tasks in a total of 1 hour per day - say 2 minutes for each message transmission, and 12 minutes for hatch opening:
From this experiment, we know we can reduce the RPi 4B power consumption to approx 40 mA while in Low Power Mode. This results in a very different daily energy consumption over a 24 hour period:
E = (1 hr x 3 watts) + (23 hrs x 5 volts x .040 Amps) = 3 + 4.6 = 7.6 watt-hours
The energy savings for this mode of operation are (7.6 ÷ 72), approx 89%, and the battery will support the RPi for
T = 720 watt-hours ÷ 7.6 watt-hours per day = 94.7 days
However, this doesn't include the 'mobile telephony internet link', or the 'electromagnet to open a small hatch'. If you'll supply those figures in an edit to your question, then we'll cover how to add the Real Time Clock, and configure your system so that it starts and stops when needed to perform a task.
Note on Battery Depth of Discharge:
For simplicity, the calculations shown above assume the battery used is capable of 100% discharge. This is impractical and/or ill-advised for virtually all batteries, and ruinous/destructive for some. In terms of your power budgets, this simply means that you must do a bit of research on the battery you are using, and determine a practical Depth of Discharge (DoD). For example, if you are using a lead-acid battery rated at 60 Amp-Hrs, and you have determined that a 60% DoD is your margin, then your Power Budget should reflect 60 x 0.6 = 36 Amp-Hrs.
Also note that the DoD margin you select is subjective - at least within a certain range of discharge - which chemistry-dependent. This discussion highlights the judgments and trade-offs that typically enter into setting a margin. This reference from "Battery University" discusses DoD from perhaps a more objective point of view.
The bottom line is that this planning and budgeting will allow you to get the "best" performance from your battery - depending on your criteria for best. Determine your DoD margin, and use that in your calculations.
