You would have to find out which agency in the State of New Jersey has jurisdiction over the state's National Guard, and send a Freedom of Information Act request to them for that person's service records. The National Personnel Records Centre in St. Louis will only have information pertaining to any time they were under federal activation. This answer is given under the assumption that '1886' in your question was a typo. The National Guard was not organised until 1903.
National Personnel Records, http://www.archives.gov
It depends how far off in paperwork was performed on you. If you are already in the guard, you will get either a med. discharge or a less than honorable for lying under your enlistment oath.
For an individual's complete military service record, you send a request to the National Personnel Records Centre in St. Louis, MO. See related link at the bottom.
For an individual's complete military service record, you send a request to the National Personnel Records Centre in St. Louis, MO. See related link at the bottom.
If you are interested in working for the National Guard the best thing to is graduate high school then type up a good resume. Take your resume to the National Guard recruitment center in your area and enlist.
When you're discharged from a Reserve component, you don't receive a DD-214 like you do on Active Duty. You will receive a simple set of orders stating you were discharged, and what the nature of your discharge was. These will often be mailed to you - particularly if you're not actually at your unit to have them handed to you - along with your 201 file and medical records. Other than that, you can request a copy of your records from the National Personnel Records Centre in St. Louis, but this route could take a considerable amount of time.
The year 1040? I do not believe there was a National Guard back then.
For information on joining the national guard visit http://www.nationalguard.com/guard-basics/faq/joining-the-guard. You can also find information at http://www.military.com/Recruiting/Content/0,13898,rcoe_guard,00.html
Go to the US Archives. If you go to the .gov website for the US Archives, you will find a link to military records. You can fill out a form online, and if it is available they will send you the records for a small fee for copying charges.
You can visit the land records office, look up the mortgage and see if a discharge has been recorded.
Start by checking any military records or documents your family might have. You can also reach out to the National Archives or the appropriate branch of the military for service records. Additionally, consider asking other family members or researching unit histories to see if you can identify which unit your grandfather served in.