When I visit Fukuoka for my business trip, I took Peach Aviation(MM), a Japanese budget airline, from Sendai(SDJ) to Fukuoka(FUK) via Sapporo(CTS).

Of course you cannot use the lounge by Japan Airline(JL) or ANA(NH), however you can use an another lounge called “EAST SIDE”.
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About lounge

EAST SIDE is located in 3rd floor in Sendai airport(SDJ), before security gate.

https://www.sendai-airport.co.jp/en/service/business-lounge/

If you hold any of the following credit cards, you can use the lounge for free of charge :

The inside is not that big.

There is a copy machine at the entrance :

You can see the fleet from the window :

You have to pay for the tinned beer.

How to connect to Wi-Fi

You can select ‘SDJEastside’ SSID.

You can receive the password at the reception.

Speed

Uploading : 10.45Mbps
Downloading : 7.88Mbps

It was a good speed, because there was almost nobody in the lounge that time.

Conclusion

I think you might have to use this lounge when you take budget airlines, it should be bigger by when lots of airlines will fly to SDJ.