I have actually commuted in a very dangerous, hilly, bike un-friendly city (Tegucigalpa, Honduras).
Here there are zero facilities for bike commuters, no racks on public transport, no bike parking anywhere and of course, no showers at workplace.
I have tackled the problem with following strategies:
Leave home with plenty of extra time. When you travel to the workplace, don't race the clock, get it at a slow pace so you don't sweat as much. It also helps that the earlier the cooler the waether. It also gives you time to cool down when you get to the office, giving you time for personal cleanup. Getting there earlier also gives you the advantage of ariving before everybody else is there, so you make less of a show off. You also get a bit more privacy in the restroom.
Re think your route: As others wisely say, you must think as a cyclist, not as a driver/bus rider. You can take a lot of advantages being a cyclist, for example, walk your bike across a park, instead of a posible dangerous ride around it. Also, for a car it's not trouble to go an extra mile to take a u-turn, on a bike, you can find a suitable pedestrian crossing (dismount if necessary). Cars don't get much trouble with up hills, but for a cyclist it may be better to take an alternative street, posibly a narower one but with less traffic, etc. as long as it is closer to horizontal.
Carry separate clothing (A shirt, as minimum), deodorant, hair gel (if you use) and a couple of hand towels. You don't need to take a complete shower, just clean your upper body with a damp towel, then dry up with the other. Reapply deodorant, hair gel and you'll be ready. To avoid odor in the office, carry a plastic bag to store the towels and the sweated shirt. (Depending on the kind of shirt you may be able to wash it in the sink and dry it with the hand dryer, so it will be ready for the ride back home.) Some advise to carry a suitcase with these items by motorized commute and leave it at the office, so you can avoid hauling it on your bike.
Some complementary strategies:
Before you risk messing up your schedule, plan a test ride. You can do it on a holyday or a weekend. Ride at the aproximate same hour so you know about the weather. (It is not the same being out there alone than inside a car). You can try several route options, analyze every hot spot so you prove what's the best alternative.
You are riding in a city that does'n know about bike conmuting, so you may be better thinking as an invisible entity. So ride in a way that you minimize chances of being hit by a car. Sometimes it's better for a cyclist to use a less crowded crossing, even if it means walking the bike for a block, or riding against the flow a few meters. You can't judge such risks from a motor vehicle, you have to bike to get it. Walking is no the same either, most of the time, walking paths are not useful for bikes due to stairs, light poles, rain grills and other "obstacles" that do not interfere with pedestrians.
An extra to point 1: When you are not in a hurry, you can turn your last 5 minutes of the ride into a low paced 10 or 15 minutes, that allow you to cool down, but, since you remain on movement, the air will dry you up before you get to the office (depending on weather of course). This helps a lot, because, most of body odor is not caused by sweat per se, but by the warm humid environment that allows for bacteria growt. If you eliminate both conditions, high temperature and high humidity in the strategic parts of your body, you can minimize odor issues.
To minimize sweating, you can also go for non obvious carrying options, i.e. avoid backpacks. Paniers and rear racks are useful, but if you don't have such, there may be alternatives. I have had success with a small messenger bag tied to my handlebars. Do not improvise too much and test before the actual commute.
Finally but most important: Safety: Do not ever ride without a helmet! You will need it the day you less expect it. Wear clothing that make you easily recognisable, use brigt contrasting colors, preferable primary colors, (Yellow, red, blue) or bright secondary (Orange, violet, and some shades of green) avoid neutral tones like gray, beige, brown. (Only one piece of clothing in these colors is enough, you don't have to look like a clown ;) ). Put reflectors in front, rear of the bike, in the spokes, in the pedals, or apropriate substitutes, like reflective ankle straps, reflective tape, etc. Nevertheless, blinking lights are even better, a white one in front, a red one in the back.
Regarding another kind of safety: do not forget a suitable bike lock.
That pretty much covers what my experience has taught. Take it with patience and practice, at least here where I live, urban biking is enough of an extreme sport so do not take a lot of risk before you really measure your possibilities. Make your commute relaxed enough so it's worthy against your other options.