Assuming the octet rule is obeyed, how many bonds and how many lone pairs will atoms of oxygen (o) and nitrogen (n) have in order to have formal charges of +1?

The Formal charge is not an actual charge of  any atom, it is the hypothetical charge if all the bonds were 100% covalent.

Oxidation state is the hypothetical charge on an atom if all the bonds are 100% ionic. There are no 100% ionic bonds.  

Formal charge (FC) calculation  

FC = VE - NBE - ½BE        [ VE=valence electrons, NBE=non-bonding electrons, BE=bonding electrons  ]

Oxygen has six valence electrons. Oxygen can have a formal charge of +1 with two lone pairs and one bond, but that doesn't give oxygen an octet of electrons.

FC = 6 - 4 - 1 = +1  

A second possibility is one lone pair and three bonds which will give oxygen a +1 formal charge. (As in H3O^+)  

Nitrogen has five valence electrons. Nitrogen can have a formal charge of +1 with one lone pair and two bonds. Nitrogen won't have an octet, either.  

FC = 5 - 2 - 2 = +1  

Another possibility is no lone pairs and four bonds.  

FC = 5 - 0 - 4 = +1