Can I deduct the expenses of a building an Airbnb project while I am living in Zimbabwe?

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When you place a property in service as a rental, you can take depreciation. This is basically "wear and tear" and is how you recover the cost of building or improving a property to be able to rent it out or use it in business.  Rental property is depreciated over 27.5 years.  The basis for depreciation is the cost you paid to buy, build or improve the property, or it's fair market value at the time it is placed in service, whichever is lower.   You don't depreciate land, because land doesn't wear out, so you have to subtract the value of the land from your depreciation basis.  You can't include the value of your own labor, only things you paid for (labor, materials, and other costs of building or improving the property like permits and installing utilities.)

It doesn't matter if the property is outside the US, the same rules apply.  If you are a US taxpayer, you owe tax on your world-wide income, and you follow US tax laws no matter where the property is.  

Also be aware that when you sell the property, you have to recapture (pay tax on) the depreciation you previously claimed.