If you are coming for work, find out whether there are special rates through the conference/event you are attending, or whether your employer's travel office has special rates with any Manhattan hotels/chains that would allow you to stay within budget. Hotels near Central Park (no view of the park) are possible (think Sheraton and Hilton, rather than hotels on the park), although your budget is limited, so even these hotels may be a bit of a stretch.

No matter where you stay, your itinerary sounds doable. Get a Metrocard and use the subways and buses, and you should be fine.