Electronic – How much resistance will I need to drop the voltage

The 1.8V is a given for the LED; if current flows through a LED it will show a more or less constant voltage across it, just like a common silicon diode will drop about 0.7V at low currents. The anode will always be 1.8V higher than the cathode for your type of LED (the actual voltage mainly depends on the LED's color). The fact that this voltage is constant allows you to calculate the appropriate current limiting series resistor:

\$ R = \dfrac{V_+ - V_{LED}}{I_{LED}} \$

or the current for a given resistor value:

\$ I_{LED} = \dfrac{V_+ - V_{LED}}{R} = \dfrac{5V -1.8V}{220 \Omega} = 15mA \$

I don't have good news for you.

Below is the discharge rate for Durcell batteries done by some company. Look at the DC label; this is a Duracell coppertop battery. The complete test can be found here: link.

There was no µA test (it would take too long), but I guess the voltage will drop below 1.5 V after about 1-2% capacity discharge. There will also be battery self-discharge (very slow). My guess estimated time for 0.3 mA (300 µA) before dropping to 1.5 V will be somewhere between 50-150 hours. You can test this; it's not that long.

I would suggest to use different kind of batteries. Alkaline and a 3.0 V requirement is just the wrong battery for this application.

You can also consider different battery chemistry. If you use some lithium AA battery, for example Energizer L91 - they have much more energy "available" before voltage drop below 1.5 V, but be careful - they have also higher initial voltage (about 1.7 V) and they are expensive (in my country they cost 2-4x more than alkaline Energizer or Duracell).

Image from Energizer Ultimate Lithium L91 datasheet: link

However if you want to pull 0.3 mA (300 µA) from battery for two years (over 17000 hours) - you need more than 5 Ah (5000 mAh) before the voltage drops below 1.5 V. This is probably too much for any AA battery available on the market.

There are also nickel-zinc AA batteries, they have nominal voltage 1.65 V, but they have less capacity (50% less than Energizer L91).

Another idea might be three batteries with a low-dropout linear regulator. Three AA batteries in series will provide more than 3.0 V until they are completely empty, but a linear regulator may be necessary for some devices - three new alkalines may have 4.95 V initially.