Yes, you could buy printed transcriptions of overtures and arias. These publications would usually be for voice and continuo (which could be just a harpsichord, or a lute, or both, and could include a cello, viola da gamba, bassoon, or some other bass instrument), solo flute/violin, or solo keyboard. They would not be terribly difficult, but we should not underestimate the proficiency of amateur musicians (because there were people who devoted A LOT of time and effort to music).
How would you get this printed music? Well, there were subscriptions. A publisher would announce a book of transcriptions/arrangements was in the works and people (mostly men at first, but more women appear in the lists of subscribers as time passed) would pay in advance to get a copy. They would get their music and there would be a page listing the names of the subscribers. This model was used for all sorts of books, and some times books that are not 100% focused on music would contain some music inside. Subscriptions were eventually done for show, just to get your name printed in the book and show you were a musical connoisseur with money to spend.
You could also buy these books in a shop without being a subscriber, or you could buy just the score for something (and/or the individual parts for each instrument). This model of selling individual works in print grew during the 18th century, and became a big thing in the next century. Printed music for domestic use eventually became a good business. Keep in mind that music would also circulate in manuscript form (people would make copies by hand).
What would these transcriptions sound like? Well, Stefan Temmingh recorded some of them (the recorder was a very popular instrument among men in 18th century England):
https://youtu.be/eLwm7hmVCxI?t=29
Here's the original aria:
https://www.youtube.com/watch?v=Nq_X1AcXwZY
Transcriptions were not just produced for people to play at home, but also were some times created for teaching. That would be for teaching how to play an instrument, basic music theory, composition.
Handel's operas were very, very popular, so there were a lot of transcriptions of his music (even after his death). Most were by other musicians, but he did write some transcriptions of his own arias.
If you want to read more about this:
https://www.tandfonline.com/doi/abs/10.1080/14723808.2019.1570752?journalCode=rrmc20
http://www.brepols.net/Pages/ShowProduct.aspx?prod_id=IS-9782503588155-1
Well we’ve got a two-parter here! The incomparable /u/erus got the second part, access to printed music scores, but I’ll take the first part!
You’ve got a bit of a misconception going - if you were an opera lover in most places of 18th century Europe (living in a city, and having good money) you could go to the opera every night, and indeed it was customary. People who had season tickets to the opera had season tickets - your ticket was good for every night of the entire season, and you could go, or not go, as you wanted. If it was a nice night, there was a star castrato on offer, and you thought the gossip was going to be hot, you bet your jangles you were going to the opera even though you’d seen it 5 times before. Premiere night was still a larger social event than the subsequent nights, but the other nights were still a social event. The performers also took into account that the people watching the opera had seen it before, and would vary their performances to keep it entertaining. So if it was during the season, and you loved opera, you went to the opera every night it performed, and that was how you got opera. (As an aside - “season” evolved over time but it was usually from the little season in fall to Christmas, a break for late winter (especially Lent, opera is verboten at lent), and the big season after lent in Spring, AKA the Carnival season in Italy. No summer opera. This is still the traditional season for opera in the West.)
For some interesting material culture check out this old post on surviving season tickets which were durable coins!
18th century diaries make lots of mention of people going to the opera on repeat without explanation, showing how normal it was.
From the diaries of Lady Mary Coke, not a super opera fan but went a lot because of social reasons:
20 December 1768 The new opera, I am told, is extremely disliked. Mr Walpole says he will go to it no more.
An opera so bad they won’t see it a second time. Delightfully terse our Lady Mary, who never bothers to actually name any operas or singers in her diaries, only people she saw in the other boxes, deeply telling of how much opera actually mattered to her consumption of opera.
From this blog post, though I was reminded of her existence from
For the florid, from the diaries of Susanna Burney, who was such an opera fanatic she would go to the rehearsals by special invitation as well as the actual performances:
Evening April 21st 1780 Ah, Ma Chere Fanni! [Her sister, the famous Frances Burney] ― I have been to another & the last Rehearsal of Rinaldo this Morning, I am returned more distractedly in love with it than ever ― yet had many little things to abate my pleasure too. My Mother, Charlotte, & I went, & were immediately shewn to a Box, wch was rather a wonder, as I never before saw so crowded a Rehearsal [...] I could dwell on the merits of every part of the Opera for ages, but think it would be making too free with your time & patience ― so shall hold back ― however I must say that even in the most inferior pieces of Rinaldo charming passages & infinite entertainment
From The journals and letters of Susan Burney: Music and society in late eighteenth-century England