Due to other medications that I take ibuprofen would be a very bad idea for me. Just check the advice in the packaging for any drug before taking it.
Get yourself covered in Deet or another "repellent" before the buggers get a chance.
As Bronski says, Ibuprofen is contra-indicated with several other drugs. Certainly it should not be taken with any other nsaids [non steroidal anti-inflammatory drugs] (for example diclofenac sodium). Also it can have an unpredictable effect on the effectiveness of anticoagulants such as warfarin (like a lot of things) which are used therapeutically for many reasons.
An anti-histamine will help to reduce redness and itching, although again watch for drug interactions (anyone on anti-coagulants needs to check with a pharmacist or doctor before taking any medication anyway)
A topical cream, such as a 1% w/v hydrocortisone will help to reduce the itching further, and aid in the reduction of swelling.
As for "manned up and let nature do its stuff", where do I start?
A swelling like that might be an allergic response to the bite. It might also be down to an infection. An infected bite can turn very dangerous, very fast. I know of two people who have died from the effects of an infected insect bite. In one case the infection moved to the bloodstream, and then the heart. The person died within days of acute infective endocarditus. If she'd gone to the doctor with the infected bite she'd still be here now.
An infection, or the possibility of one involving lots of swelling, requires treatment by a qualified medic. A quick dose of antibiotics could save you a whole world of pain and loss.
I lost my commercial pilots licence following a blood infection. It's not something that you can "man up and get through". Untreated, a blood infection is almost 100% fatal!
Sorry for the doom and gloom, and I'm glad you got away with it, but I'd always recommend seeking medical advice for such symptoms.