How should I cook a 1lb bottom round roast beef in the oven?

Given your lack of tools, we're going to have to get MacGyver on this roast.

• You'll need some sort of flavorful liquid (or a combination of them). Chicken stock or beef stock would work. I'd normally also recommend using something acidic like red wine or tomatoes, however, using this method (which is based on tinfoil), acids can react with the aluminum to produce off flavors. If you want to cook with wine or tomatoes, you can transfer the meat to your casserole after you brown it. (More on this below.)
• You'll also need some type of mild tasting oil. Vegetable, corn, canola/rapeseed, peanut, light olive oil, will all work.
• Salt.

That's it in terms of software.

Hardware:

• Paper towels.
• Tinfoil (preferably the big, wide kind, and heavy duty).
• An oven.

That's it.

Algorithm:

This is basically the same technique that Cos Callis proposed, just using different tools.

Adjust your oven's rack to about 6 inches from the broiler. Turn on your oven's broiler on its highest setting.

Lay out two layers of tinfoil on your counter. Make sure the tinfoil is big enough such that if you were to place the roast in the middle of it there would be a border of tinfoil around the roast that is at least as wide as the roast itself.

Optional: Put the tinfoil on a large sheetpan or cookie sheet. This will help you in transporting the device to/from the oven, and it will also ensure against spillage in your oven. If you do things right you won't have to wash the sheet.

Pat the meat as dry as you can get it and put it in the center of the tinfoil.

Pour a bit of oil on top of the roast and rub it all over. You only need enough to barely coat the entire roast.

Generously salt both sides of the roast. I prefer using kosher salt, because it is less saline by volume, so you can more evenly distribute it without over-salting. You'll probably use a good tablespoon or so.

Put the roast under the broiler. Wait about 5 minutes, but that number isn't exact. Use your eyes. Just wait until it looks good and brown and crusty and delicious. Just don't blacken the entire thing.

When the top side is browned, flip it over to the other side. The best tool for the job here is a pair of tongs, but a fork will do.

Brown the second side.

Remove the roast (which is hopefully on top of some type of pan for your convenience).

Turn off your broiler and reset your oven to 275 degrees F.

Fold up the sides of the tinfoil around the roast to make a sort of pan around it. Try to leave as little room around the roast as possible.

Alternatively, if you want to use acidic ingredients, you can transfer the roast to the casserole at this point. The reason why you couldn't use your casserole from scratch is that it is probably not broiler-safe.

Pour in enough of the beef/chicken stock to just barely cover the roast. It shouldn't require very much liquid.

Once the liquid is in, fold the tinfoil on itself to seal the top, such that you end up with a tinfoil package containing the browned meat and the stock. Try and make it airtight.

Stick that in the 275 F oven.

Just let it sit in there for ~3 hours. You don't need to touch it.

After ~3 hours, take it out. Hopefully the tinfoil didn't leak.

If you did this correctly, all the cleanup you need to do is crumple up the foil and throw it out.

You can also experiment with adding other vegetables (e.g., thinly sliced onions or whole garlic cloves). You can even add them from the very beginning.

As I mentioned in a comment on Cos Callis's answer, bottom round roasts have a lot of tough connective tissue which needs to be cooked low and slow (<300F) in the oven in order to become soft and tender. However, if you were to cook the beef the entire way at that temperature the meat would end up being gray and tasteless. The reason is that the Maillard reactions (which are what turn the meat golden brown and make it taste more meaty and delicious) don't really occur below ~300F, which is the purpose of the initial sear.

Do you have a crock pot? If yes this is the ideal tool for this. If not, you want to try to impersonate a crock pot with your oven, low, slow and with moisture.

Start rubbing the roast with olive oil and then some salt and pepper and rub that in. Then in a hot skillet sear every surface (hold it with tongs if you need to in order to get a good all around sear) just till it has a bit of a crust. Once the searing is complete put it in your crock pot with some potatoes, carrots, celery and about 1 inch and let it cook on low for several hours in a crock, maybe less in an oven set to about 250F. When the roast reaches an internal temp of 160F-165F it is done. There are a lot of ways to expand on this, but as a basic technique it is a good starting place.

[edit]: @derobert: I don't know how I missed that OP says she doesn't have a crock pot. The casserole dish with lid should work well as a substitute. And "Yes" garlic & onion would make excellent additions to the basic list I provided. I was more trying to steer OP to "sear then slow cook with liquid and veggies..." as a technique rather than a specific recipe.

Simple.

Take slices of fresh garlic, make a slits in the meat and shove the garlic in there. By slit, I mean take a sharp small knife (paring maybe), and jab straight into close to halfway.

Salt, pepper, rosemary, crushed beef boullion. Mix together and rub that stuff all over the outside. Put it in a pan with a little water and cook at 400 until done.