The material with which you cover up a tattoo on your arm depends on whether the tattoo is healed or not. If the tattoo is not fully healed, you should only cover it when absolutely necessary -- because tattoos need a healthy exposure to air in order to heal properly -- but if it is healed, you can cover it at any time. Whether healed or not, the colour of the material must be darker than the tattoo or at least thick enough to not be transparent.
Wash your tattoo with warm water and lather protective ointment or hand lotion over the area.
Put on a loose, dark shirt that will cover the tattoo completely while still allowing it to breath and heal.
Pull the fabric of the shirt away from the tattoo any time it sticks to the lotion or ointment and try to clean and reapply every four hours or so, if possible.
Take off the shirt as soon as you can and allow your tattoo to heal, uncovered.
Put on a shirt that adequately covers the entire tattoo on your arm and is dark or thick enough to not be transparent. A shirt may be the best way to cover an arm tattoo, but it is by no means the only way.
Pull a tennis sweat band over the tattoo if it is small enough to fit underneath and if it is appropriate for your situation.
Wind an ace bandage around the tattoo if the design is too large for a sweat band and you don't want to wear a long sleeve shirt. Some people may ask about your injury, so you can either make up a story or tell them the truth.
Stick a band-aid over the tattoo if it is extremely small. Most people wont give a band-aid a second glance since they are so common.
Never bandage, bind or wrap a fresh tattoo in any way until it completely heals. Proper exposure to air is critical for the healing process.