If you want to do what Japanese people would do, you take money in a special envelope to the otsuya or the funeral itself (some people go to both). You won't receive any invitation to the funeral. You just go unless it's made very clear that the funeral is for the close family only (somewhat unusual unless there is a separate service for non-family members later).

The going rate for the amount of money depends on the relationship between the person (or company) giving the money and the deceased (or more likely the surviving family member(s) of the deceased). When my brother died a few years ago individuals from his company younger than him brought 3,000~5,000 yen each, many of those in the same generation as my brother or older (or more senior in the company) brought 10,000 yen. Departments within his company and other companies he had business contact with brought 20,000~50,000 yen and his company sent 410,000 yen which I thought was a strange figure (numbers 4 and 9 are considered unlucky in Japan and should be avoided even for funerals). There must have been some going rate within the company based on the length of service.)

Japanese funerals are very expensive and monetary gifts are supposed to help towards the cost. Especially after the otsuya food and drinks are offered and although (un)fortunately I missed it since I couldn't be there in time my brother's otsuya apparently ended up as a company drinking party!

My idea of Japanese funerals is better represented by this film.
http://www.imdb.com/title/tt0089746/