As far as I know, one of the most economical flights is a roundtrip to Kansai from your city. It could depend on the exact days, the season, and the arrival and departure times, though.
Not only the price but also your time usable in Japan is determined by the arrival and the departure times and the season; the daylight hours.
Usually, people feel enough if you stay two hours in a spot. Padding one hour for a move to the next,
three hours per spot is a good course to consider. If you start from 9 AM, you could only do one spot in the morning. After lunch, you could do one more. If in winter, that is the end. But in summer, you could do three in total.
Even without daylight, you could do some kind of night spots, such as Shinjuku, Shibuya, Gion,...
Even before 9 AM, there are spots for such hours, Meiji Shrine, Fushimi Inari, Itsukushima Shrine,...
Are you OK in such hours? I usually use morning hours for transportation. Could you follow such strategy?
So, I already said there are no answers in general. Even there are no must-sees for a first trip. Some do just for ski, or flowers in Hokkaido as a first trip, skipping both Tokyo and Kyoto.
Anyway, here is one of the most conservative itineraries for your ordering. It looks not too bad. If you are satisfied with it, just take it.
D1 A late flight to Narita. Just stay in Shinjuku.
D2 Shinjuku 915-(bus)-1100 Kawaguchiko (5h) 1640-1825 Shinjuku
The sunset is around 1630 in winter.
http://bus-en.fujikyu.co.jp/highway/detail/id/1/#timetableD3 AM: Asakusa, PM: Shibuya
D4 Shinjuku 901-(JR)-1148 Kyoto, PM: Kiyomizudera
D5 AM: Fushimi Inari, PM: Arashiyama
D6 Kyoto 900-(Kintetsu)-935 Kintetsu-Nara (7h) 1700-(Kintetsu)-1735 Kyoto
https://world.jorudan.co.jp/mln/en/?p=0&from=Kyoto&to=Nara&date=12%2F2...https://world.jorudan.co.jp/mln/en/?p=0&from=Nara&to=Kyoto&date=12%2F2...D7 Kyoto 900-(trains)-1130 Koyasan
https://world.jorudan.co.jp/mln/en/?p=0&from=Kyoto&to=Koyasan&date=12%...D8 Koyasan 908-(trains)-1343 Miyajimaguchi
https://world.jorudan.co.jp/mln/en/?p=0&from=Koyasan&to=Miyajimaguchi&...D9 AM: Mt Misen, PM: Hiroshima
D10 Miyajimaguchi 900-(JR)-1353 Tokyo, PM: Ginza
D11 An early flight from Narita.