Well technically, yes.
You could say your current 1280x1024 resolution is better because it has more total pixels than either 1280x720 or 1366x768. It's also only slightly less than the total pixels in 1600x900.
If saving money is really important here, the 1600x900 is going to give you very similar FPS to what you get with your current 1280x1024. The only advantage is the slightly extra width in gaming with the 16:9 aspect ratio, which is standard these days.
Going for 1080p, however, is going to feel much more like an upgrade and will give you more versatility in the future. You can experiment between 720p, 900p and 1080p at will for the best mix of clarity/FPS, depending on the game. If you do choose to buy a discrete GPU later, you may not need to upgrade your monitor a second time if you're happy with it as is.
On a budget I would always consider the option of a used monitor! This might net you a better monitor for the same price as a new one. I'm not sure how tight your budget is and used isn't everyone's thing, though.
What are the monitor choices and what are the differences in pricing? It will be easier to help that way. Links and your budget will be best, if you have them.