Hello Jake
All of that info is good, but you need to tell Mr. Farmer. Also, I think its crucial that you figure out the visa situation, as it will be more complicated than wwoofing, and you can't expect Mr. Farmer to arrange a visa for you. As I mentioned, internships are not part of Japanese life, so there is no "intern visa". A tourist visa only allows you to stay for 3 months (perhaps longer if you extend, but there is some risk that you will not be granted an extension). If you get paid, it is a further complication as there are visa and tax consequences.
So, I'll take a stab at the opening part of the letter, but I think your internship request will require some explanation, and perhaps a much longer, technical letter.
....さん、お久しぶりです!
Fine - maybe leave off the exclamation point unless you two are cracking good friends
元気ですか?
Fine - maybe put お before 元気, like you did above with 久しぶり
去年は(his farm)でウーフしました。私のことを覚えていますか。
Consolidate this into one sentence, and put your name into it e.g. 去年、Mr. Farmerさんの農場でウーフしましたジェークサムズです、私のことを覚えていますか?
今は私はオランダの大学でオーガニック農業を勉強します。大学の名前は(my school)です。
Again, consolidate this a bit 今はオランダの(my school)大学でオーガニック農業を勉強しています。
大学一年生です。
Leave off the 大学 since you've already said you are attending university.
The intern part I will leave you to take a stab at.
What is it?
How long is it? From when to when?
What you need him to provide you with?