Cheapest transport from Akasaka to Narita Airport (2 pax)

The cheapest option will almost certainly be taking the Marunouchi Line subway from Akasaka-Mitsuke to Ginza (Y170) and transferring to the Access Narita bus to Narita (Y1000). Or you can also take the subway an extra stop to Tokyo stn, which is also served by Access Narita and has twice as many departures.

Note that IC cards are not valid for the bus, you'll need to pay cash as you board or buy online in advance by credit card.

The Keisei Limited Express departs Ueno and Nippori and costs 1030 Yen with an IC card. This journey will take about 70min.

You can get to Ueno from Akasaka on the Ginza line in about 25min for 200 Yen.

This is a good, cheap and fast option if you want to avoid buses.

With an IC card you can go from Akasaka to Machiya on the Chiyoda subway line (195 yen), and from Machiya to the airport via Keisei (973 yen). This is three yen more expensive than @jpatokal's answer, and slightly more inconvenient as there are two train transfers, instead of relaxing on the bus. But, this weekend may see some highway congestion due to the 5-day weekend, so you may alleviate some anxiety by taking the train and bypassing the potential traffic jams.