I know that many people prefer to hose their mowers down with a garden hose. I admit I have done it. Still doing that is risky. If water gets into the wiring air filter or engine you could really do some damage. Hand washing with damp rag is a drag but also your safest bet.
The Sun Joe MJ104E has more than 1250 five-star reviews on Amazon. Buyer Jim has had his Mow Joe for more than a year and left it in the weather and abused it otherwise. He is happy with how it has performed. He cautions against buying the 16-gauge extension cord recommended by Amazon. Instead choose 12 gauge for 100 feet or more and 14 gauge for under 100 feet.
Place a drip pan or other container on the side of the mower with the filler. Carefully tilt the mower so oil drains out and into the pan. Discard the spent oil into a proper disposal vessel. Now slowly pour fresh oil into the mower. For the specific type of oil you need check your instruction manual. Make sure to add only as much as your particular model requires. Remember to let the mower sit undisturbed for a few minutes so the oil settles properly inside the engine.