So, if all goes according to plan (yeah, I know about plans), I’ll have one day and night in Montreal, a monday, the first week of June. My interests would be parks, museums, historical photography, an excellent meal (preferably something regional), and some music (jazz?) that night. I know nothing about Montreal, so help me out,
thanks.
Oh yeah, and a decent cheap hotel
Auberge de La Fontaine. 2005 was the last time we were in Montreal, we stayed there and we loved it. Lots of decent restaurants/bars in the area and seemed like every other one had music on at least some evenings.
Go over to Oratoire St Joseph, take pictures, enjoy the view. Tour the top of Mount Royal (enjoy the view, take pictures). Come down the hill on Parc Avenue, make your way to either the Fine Arts Museum, McCord Museum (both on Sherbrooke street downtown), or Pointe à Callière (my recommendation); Pointe à Callière is in the Old Port, so lots of lovely buildings, views and things to take pictures of. I think you’ll love the Old Port, based on your OP.
You can pick a random restaurant (they are pretty much all good) or make your way back up the hill to the Plateau and grab a poutine at La Banquise on the corner of the Lafontaine Park (on Rachel street). The Park is very lovely to explore, so also great for photos. La Banquise is open 24/7 so you have absolutely no excuse to miss it other than lack of interest! They even have a 100% vegetarian option, if that’s your thing.
Someone else can give better jazz recommendations, but there’s likely something going on at the Old Port or on St. Denis. You’re too early for the Jazz festival (unless you move your trip to the first week of July!) but there’s always stuff happening.
I can’t really recommend cheap hotels, as I live here and never need one, but the Auberge de Lafontaine is rather pretty and close to the Lafontaine Park, so good for exploring. My mom often stays at the Chateau L’Argoat on Sherbrooke near St. Denis when she is in town for conferences and she loves it (if a public school board can afford it, I’m sure it’s decent!) I’ve seen one room there and it’s rather charming and just steps from the metro (Sherbrooke Station, Orange line) and shopping/restaurants on St Denis and the Main (St-Laurent Blvd).
All of these things are long walks from one another (doable, if you like to stroll, but harder with your time limits), but definitely easy metro and/or bus. You can get day passes for the transit system, or it’s $3.00 per ride (yeah, it’s pricey, but the monthly pass is quite cheap for those of us who live here).
Feel free to ask more questions; I can give you links if you want some…to be honest, I’m feeling tired and lazy at the moment, which if why I’m not giving you any 
I, too, will be visiting for a week on business in July and will extend an extra day or two to see some sights. Like the OP I know very little of the city so if you’ve been there then please share your suggestions. Thanks in advance.
Thanks, mnemosyne and TheMightyAtlas, for your suggestions so far.
I was in Montreal about 5 years ago and my frien and I stayed in a B&B in the Gay Village - awesome location, super close to the metro and super cheap. The one we stayed in has been sold (it was featured on Buy Me, actually) but there are loads around there.
I have been to Montreal several times in my life and enjoyed it every time.
I haven’t been there since 2004, though, and for several years I’ve intended to make a trip there to visit the restaurant Au Pied de Cochon. It’s not affiliated with the fairly well known one in Paris in Les Halles, instead they add some French Canadian aspects to haute cuisine, and I’d love to try their Foie gras poutine (yes, the humble local comfort food of “fries with cheese curds and gravy” amped up with high grade foie gras) and Canard en conserve (“Duck In A Can”).
Thank you all for the recommendations, but it looks like a plan is just a list of things that don’t happen this time around.
The Canadian Grand Prix is 8-10 June. I think you are going to find cheap accommodation rather scarce.