I think in reality what the OP wants to do is to improve her body composition (the ratio of fat to muscle she carries) as opposed to flat out building muscle and bulking. Her bingo wings may be a result of losing skin elasticity (and therefore surgery will probably secure the best outcome.) Alternatively it may simply be fat deposits which will reduce over time.
In the circumstances the key things here will be a consistent caloric deficit, adequate protein (though it doesn't have to be ludicrous levels) and some form of resistance training for her upper body (the cycling particularly if she incorporates some hills / intervals at some point will be more than enough to deal with her lower body given her stated goals. If she incorporates resistance training focusing on her lower body like squats, weighted lunges etc the likelihood is she will overwork and tap into her recovery ability.)
So:
1) Keep the quality of your diet high by eating lots of fresh vegetables (particularly green leafy vegetables) moderate amounts of fruit, starches (like potatoes / rice) and lean protein. Have treats and alchohol in moderation. If you aren't losing body fat over time then reduce portion sizes or consider calorie counting.
2) Keep up the cycling
3) Incorporate 2 - 3 short upper body workouts per week centre around: push ups, assisted pull ups, assisted chin ups, tricep dips, planks etc. Alternatively find a decent kettle bell routine. If you like using barbells and weights do that but it is by no means necessary (what is important is progressive overload of muscles which trigger hypertrophy...)
4) Be consistent
5) Be patient
6) Profit