How effective is a shield component during the first round?
Shields are not very effective during the first round.
First, it should be noted that the negative effects of the Pirate, and the Combat Zone adventure cards are some what avoidable. In the Pirate encounter case, if you make sure that your ship has 5+ Laser power, you will not be shot at. With the Combat Zone, you just need to make sure that you don't have the least Laser power.
Second, you need to figure out the odds of drawing that particular Adventure Card. The chances that you will actually draw a particular adventure card during the first round is 8/20 = 40%, if you don't look at the Adventure cards. Since a single shield component can either face up or down. The majority of the small laser fire, and small meteors come from the front, so you should probably orient the shield to face forward. If you do place a shield component, and you happen to draw the Pirate encounter and you don't have the required 5 Laser power, your shield will only be effective 24/36 = 66.7% of the time for each small laser (since rolling 2-4 or 10-12 misses your ship anyway). This means, that if you don't look at any of the adventure cards, you are only going to effectively block:

Block 0 Small Shots: (60%) not drawn + (40%) x (12/36) x (12/36) = 64.4%
Block 1 Small Shots: (40%) x 2 x (12/36) x (24/36) = 17.8%
Block 2 Small Shots: (40%) x (24/36) x (24/36) = 17.8%

                ___
11.1% 5     ___|   |___
13.9% 6 ___|    ___    |___
16.7% 7|       |___|       |
13.9% 8|        ___        |
11.1% 9|___ ___|   |___ ___|
         5   6   7   8   9
        11% 14% 17% 14% 11%

Nearly 65% of the time, it wouldn't matter if you bought a shield or not and only 18% of the time you would have prevented 1 lost component, and 18% of the time you would prevent 2 lost components. This isn't a completely accurate though. It depends a player in-front of you defeats the Pirate encounter for 4 Credits, or you draw the Pirate adventure card before drawing one of the other Meteor Cards. If you draw the Meteor Cards after you draw the Pirate encounter, then your shield is more likely to be effective, but not that much more effective.
For example, if you where shot from the front, and later draw any of the Meteor cards, you have an increased chance of losing another component because of additional exposed connectors. The increased chances are only the chances that you roll the same column/row of the component that got blasted from the front. For example, if lasers destroyed 7 &amp; 8 from the front (Note: Getting hit 2 or more times from the Purple encounter happens about 30% of the time), then if you draw the Lss (TLR) Meteor card:

Small Meteor No Hit: (26/36) x (26/36) = 52.2%
Small Meteor Hit 1x 6/8: 2 x (10/36) x (26/36) = 40.1%
Small Meteor Hit 2x 6/8: (10/36) x (10/36) = 7.72%

You only have a another 50% chance of getting hit again by a small meteor one or more times. I won't work out the exact math, but realize that the odds of losing extra pieces from small meteors is fairly small (60% of the time it won't be in the adventure deck. When it is, only half the time you will draw the Meteor card after the Pirate encounter, and finally even if you do draw the Pirate encounter first only half that time will you be hit by a small meteor hitting an exposed connector).
If you do look at the adventure cards, and neither of the cards are in the 6 cards, you only have an additional 14.3% chance that the Pirate card will be in the remaining 2 Adventure cards.
In general, you should probably avoid placing a Shield component. The only exception would be if you saw the Pirate encounter or the Combat Zone card, and you were unable to meet the requirements of 5+ Laser or not the least laser power.