No one here can tell you what you can get with one months study. I know people who took the GMAT without even knowing what material was tested and got a 700+.
One friend of mine took it "for fun" and scored a 780. He just wanted to see what he would get. Wasnt even interested in grad school at that point. He didnt even know the format.
Another friend of mine studied on the train during his trip to the exam, for about a half hour, scored a 720.
Another friend of mine took a 9 week prep course, studied at least 30 hours a week, for all 9 weeks, and scored a 540. He then studied another 120 hours on top of that and pulled a 680. In the end, he put in well over 400 hours.
One girl I know studied for seven months and got a 720.
I studied for about 15 weeks, averaging 25 hours a week for 9 weeks, (225 hours), scored a 640, and then put in another 6 weeks pushing 30 hours a week, (180 hours) before I scored a 730. All told, I put in about 400-500 hours of study over a period of 15 weeks. I took somewhere in the neighboorhood of 20 full length practice exams. That's about 80 hours of exams. My story is here:
https://www.gmatclub.com/phpbb/viewtopic.php?t=30030
My advice?
If you target score is 620 - take a practice exam and see how you do. If you are well above that number, take it. If you are scoring 630's, 640's, 650's.... probably wait. My friend who scored a 540 on his first attempt was scoring 650's on practice exams. When I got my second 640, I was scoring 720's on exams. Only on the third exam was I able to match my practice performance.
Also, yes
Manhattan GMAT - get these books, especially the SC one
I also highly recommend the class, but if you only have a month to do this in, that might not work.