Is it cheaper to fly on a specific day of the week?

Yes. Mid-week flights (Tuesday or Wednesday) are consistently 10-15% cheaper than weekend departures.

Should I buy a round-trip or two one-way tickets?

Always book a single round-trip ticket. Two one-way tickets on different airlines are almost always more expensive and make you liable for re-booking both if one leg is cancelled.

Booking Flights to Japan: The Practical Guide

Book your flights 3 to 4 months in advance to secure the best balance of price and flight duration. Use a combination of Google Flights for price tracking and airline-direct websites for booking to ensure you have control over your reservation and seat selection.

Set up a Google Flights tracker. Enter your departure city and destination (NRT or HND for Tokyo, KIX for Osaka). Toggle 'Track prices' on. You will get an email notification when prices drop significantly, which is usually the best indicator to pull the trigger.
Choose your airport wisely. Haneda (HND) is much closer to central Tokyo (20-30 minutes) than Narita (NRT), which can take 60-90 minutes. If the price difference is less than $100, always prioritize a flight into Haneda to save yourself time and expensive airport express train fares.
Book directly with the airline. Use search engines to find the deal, but book on the airline's official site. If a flight is delayed or cancelled, third-party sites (OTAs) are notoriously difficult to deal with; airlines will prioritize customers who booked directly.
Check the 'Multi-city' option. Don't force a round-trip if your itinerary is linear. Book an 'Open Jaw' ticket (e.g., fly into Tokyo, fly out of Osaka/Kansai). It saves you the cost and half a day of travel time returning to your original arrival city.

Is it cheaper to fly on a specific day of the week?: Yes. Mid-week flights (Tuesday or Wednesday) are consistently 10-15% cheaper than weekend departures.
Should I buy a round-trip or two one-way tickets?: Always book a single round-trip ticket. Two one-way tickets on different airlines are almost always more expensive and make you liable for re-booking both if one leg is cancelled.