I don't know how to say this other than direct. Your budget is unrealistic. I'm thinking you'll need to at least double, if not triple, that. Yes, there are lawn tractors that can be had for that $1k mark, but I will not endorse them for I would not buy one.
With regard to mower type, unless you have the need to perform a lot of ground engaging, landscaping activities (something I wouldn't do with anything less than a garden tractor given the size of your property), you simply cannot beat a ZTR when it somes to mowing speed. In fact, if you did get one, you'll be wondering what to do with all of your spare time.
However, and again this goes directly to your budget limitations, even the cheapest homeowner unit I see at Home Depot is x2 what you want to spend.
Ariens Zoom 34 in. 14.5 HP Briggs & Stratton Hydrostatic Zero-Turn Riding Mower - 915157 at The Home Depot
I know nothing about this model, and quite frankly, most likely would not consider any entry level ztr.
If you don't mind continuing to walk, then a similar sized unit is still x0.5 more than you're willing to go.
Ariens 34 in. Self-Propelled Wide Area Walk Behind-Mower - 911413 at The Home Depot
Again, I know nothing about this machine.
Some may suggest a used commercial unit, and that may be a possibility depending upon where you live, but I've never seen a commercial anything at a dealer around here priced at less than $2500, and then they've all been high hour (2500+) units.
Just for reference, my property is mostly flat 1.5 acres with about 10 mature trees. Four years ago, I purchased a new Dixie Chopper Silver Eagle LT2700-60D (yes, that's 27 hp with a 60" deck). At that time, this unit was $6500; today it's over $8k. If I were to buy today, it would be a toss-up between a Husqvarna MZ6128ZT, or an Ariens Max Zoom 60 (also known as a Gravely ZT 60 HD). Any of these can be purchased for $5k. I might consider a 52" - 54" deck, but with only $500 difference between the two, why bother?
HUSQVARNA MZ6128ZT - Zero turn mowers
Zero Turn Lawn Mowers, Ariens® Max Zoom 60 Product Overview
Zero Turn Mower - Gravely ZT HD