In a short letter, should I mention what I am going to be explaining in the next section, at the end of each section? I am writing a 3-page letter. Space is limited, so I am trying to find out whether I can eliminate additional words. 

I noticed in most academic paper, the introduction ends with explaining whats going to come in the methods, results and discussion. 

Then sometimes at the beginning of the methods section, some authors (not all) state whats going to come in the methods section and then goes into more details. 

At the end of the methods section or beginning of the results section, some papers go into what’s going to come in the results section and then goes into details.

Similarly, for the discussion section.

I understand from reading online, that’s it’s a good idea to give an idea of whats going come in the methods, results and discussion sections at the end of the introduction 
(http://abacus.bates.edu/~ganderso/biology/resources/writing/HTWsections.html#introduction).

However, is it customary to summarize each section (something like a mini-abstract) at the beginning of each section/end of the previous section?
You want your reader to be able to figure out what's in the paper without reading through all the technical details, and then going through the trouble of synthesizing what you've written to figure out the what the paper means. If you want your paper to have any impact at all, you need to put an executive summary somewhere in it. 

Whether you put all the details of the summary at the end of the introduction, or put a less detailed summary in the introduction, and expand on the details at the beginning of each section may depend on the author and/or the field. It may be that in your field, people expect summaries at the end/beginning of each section, and will look for them there.

But if you're careful about what you say and how you say it, you don't need to repeat details from the introduction at the beginnings of the section, so putting the summaries at the beginning/end of the sections shouldn't use much extra space.