What were you hoping to study for your graduate degree in the US? The only one year masters degree I have heard up are along the lines of sports management as I have had some friends do this at the University of Texas (TJ) and Oregon (XC/5k). However, they did this immediately after completing their undergraduate degree.
I do not think you would be allowed to do Division 1 athletics, but you might be able to do Division 2 or 3. The Division 1 athletic programs (i.e. all sports) have a set timeframe of 5 years after you first enroll in the school. Division 2 and 3 I believe are not as strict and use terms of number of semesters...
As for if you're eligible, I do not see anything on the NCAA website that discusses international graduate students, but they do have a phone number you could contact if you want to ask them directly:
Graduate Student Eligibility
A student-athlete who is enrolled in a graduate or professional school of the college or university which he or she previously attended as an undergraduate student may participate in intercollegiate athletics, provided he or she has eligibility remaining and is within five calendar years of initial full-time collegiate enrollment for Division I and within the first 10 full-time semesters of collegiate enrollment for Divisions II and III.
Also, a student-athlete who has eligibility remaining and is within the specified 10-semester period may participate while enrolled in a graduate or professional school at a Division II college or university other than the institution at which he or she completed an undergraduate degree.
Finally, a student-athlete who has eligibility remaining and is within the specified five-year period may participate while enrolled in a graduate or professional school at a Division I college or university other than the institution at which he or she completed an undergraduate degree, provided he or she meets the criteria of the one-time transfer exception to the general transfer residence requirement. That exception is as follows:
•The student-athlete must be seeking to participate in a sport other than baseball, football, basketball and men's ice hockey, except that a student-athlete who seeks to participate in NCAA Football Championship Subdivision may use this exception only if transferring from a Football Bowl Subdivision program.
•The student-athlete may not have transferred previously from another four-year institution unless he or she transferred previously and received an exception to the transfer residence requirement because his or her institution either discontinued the sport or did not sponsor the sport in which the student-athlete is a participant.
•The student must have been in good academic standing and eligible to compete had he or she decided to remain at the previous institution.
•The student-athlete's previous institution must certify in writing that it has no objection to the student-athlete being granted an exception to the transfer residence requirement.
NOTE: This exception does not apply to a student-athlete who attends a Division III institution for graduate school, unless the student-athlete is attending the same institution at which he or she was an undergraduate.
NOTE: If the student-athlete transfers to the certifying institution from a Division III member institution and meets the above-mentioned conditions, he or she may be eligible to compete but may not receive athletically related financial aid during that year.
Please contact the certifying institution or conference for additional information. You also may contact the NCAA academic and membership affairs staff at 317/917-6222 if you have further questions.