Are you sure your Flexinol is 0.001 inches in diameter? That would use 20 mA, not 1000 mA. The 1000 mA stuff is 0.010 inches.
http://www.robotshop.com/ca/PDF/flexinol-technical-data.pdf
If you DO have 0.001" Flexinol which needs only 20 mA then you could drive it directly from an Arduino pin. You will need 250 Ohms of total resistance (at 5V, 5/0.020). The Flexinol is 45 ohms per inch so if you used, say, a three inch length of Flexinol that would be 135 Ohms (45 * 3). To get the total to 250 that would call for a 115 Ohm resistor. The resistor would be taking about half the voltage (115/250=0.46) or 2.5V. Two and a half volts at 20 mA is 0.05 Watts so even a tiny 1/8W (0.125W) resistor will be big enough.