This may be the blind leading the blind, but my guess?
"Ultimo" means last as in "last cookie in the jar."
"Pasada" means last as in "last week."
They both have cognates in English to help you remember them. Pasada is like passed -- the week has passed. Ultimo is like ultimate -- the ultimate cookie. We also have the word penultimate in English, which means second to last.
This sentence is literally “this past week" as in the one that occurred before this one. Most English speakers use “last week" to describe this. 'Última semana' would refer to the final week; there will not be another week. So you don't actually say 'last week' in Spanish, because that would actually mean it is the end of the line, not 'that one which occurred before this current one.'
likewise where the blind leading the blind i take a guess also. i believe that "pasada" like "estación" is time specific eg last night,and the winter mostly, but ultimo like temperado refers to things last mango or mango season. please correct me if i am not right for i am here to learn
'Semana' always needs another word. This is usually an article. Sometimes the article is unnecessary if certain descriptive words are used which pertain to a specific semana:esta, una, terced, otra...
This sentence in Spanish is literally “the past week". It is translated to “last week" because that is what English speakers say in most situations. As you can see, the article appears in English too. But most of us prefer “last week", except when using specific sentence structures.
In most cases, last week would be used by native speakers. However, the week past is acceptable among English speaking folks. I got it wrong for using the week past. Sorry Duolingo, I live in an English speaking country and could say to another English speaking person "the week past" right now and I would get no strange looks indicating "what!?"