For public HS there used to be a monthly fee of about 10,000 if you were above a certain income, possibly they abolished it. Otherwise - similar to public MS - besides the school uniform, the sports uniform, the sports club uniform (if applicable) and one school trip (this time abroad), and where applicable - many public HS allow bicycles - commuting fees (student teiki have a reduced fee), no additional costs apply.
You normally have to prepare for juku costs though, very few students go without, as here you will not only study the exam contents, but the most efficient strategy how to pass the entrance tests and get used to the exam set up via many mock tests. Most students start at least two years before graduation.
Monthly cost can be around 20,000 to 30,000 Yen or more (you pay per subject: Science, Japanese, English mainly), plus a summer course and a winter course (optional but they juku heads are often quite experienced Sales people very good at exploiting our fears
).
You can only sit the exam for one public HS, but for as many private schools as you like (application fees required). Typically those students who want to enter a public HS, aim for one a bit below their level and as a back-up for one or two private schools, where entrance is almost guaranteed (considering the student‘s level).
A bit off topic: Although the fact that jukus exist/have to exist is an expensive nuisance, they are (from my limited experience), much better at finding the right and realistic fit for your child than the MS teachers, who are not only extremely overworked, but often not so familiar with the schools in the vicinity, as they may be recent transfers from other areas. Also, the jukus regularly evaluate (you will be presented with detailed computer prints several times per year) their students and are very good at judging for which schools (at that stage ) they will have a high chance, which was also great for foreign parents like we both are, who had little insight of how the system works - at least in the beginning.
Our private university cost about 1.4 Mio per year, plus entrance fee (300,000?). For the (privately organized) year abroad, we had to pay around 150,000 to secure the place.
Again juku fees (for most) apply about two years in advance and the quite costly application fees of a few man per university your child wants to sit the exam for - about five in our case. Also, depending on the exam sequence, it can happen that you will have to pay a fee to secure the Uni of e. g. your 2nd choice, in case you cannot get into the one of your first choice (where the exam was unfortunately later), way for the schools to generate additional income.
Good luck!
If I remember right it might have been about ¥30k/month each. (neither went to juku, so 'saved' some money there)