How long can I wait with spider bite to see if it goes away without seeing a doctor?

Spider bite. Generally you can wait for several days or a week with an uncomplicated spider bite, by which time it will mostly have healed up. There is no specific treatment for most spider bites, and you just want to clean the wound and keep it clean and keep an eye out for infection or other complications.
Depends. If the area becomes red, warm, swollen and painful, then you need to see a doctor. Otherwise, if this is not occurring and there is no expanding necrotic (black dead tissue) area, then simple wound care is probably sufficient.