Agree that you need to contact a mortgage lender and have them verify employment, salary history, credit history, savings, etc. You need to request a pre approval - a letter that states the lender is committed to giving you a mortgage for a fixed amount. Real estate agents need this letter before they will show you any apartments.
Next, be aware that agents are either licensed in NY or NJ. A few are licensed in both states but very few. Hoboken will give you more space for your money and will be cheaper, but you have to factor in transportation costs if you work in Manhattan. You should keep your options open and explore other towns in NJ besides Hoboken. You may need to buy a car living in NJ - but do this AFTER you buy your 1 bedroom apartment. Mortgage lenders hate buyers who make a major purchase before they close on the house.
Another option besides Manhattan is Queens or Staten Island. Both are much cheaper than Manhattan.