Probability that sheepdog performs at least one task successfuly - Am I doing this problem right?

The complement of at least $1$ is not at most $1$.

The complement is if none of the task is perform.

Hence just compute $$1-\prod_{i=1}^4 (1-p_i)$$

By sheepdog, I assume you mean 'good sheepdog' as defined in the question. Let $X$ denote the number of things in the list satisfied. If you want the probability the at least one of the 4things listed is satisfied $P(X\geq 1)$, you work it out by solving $ 1-P(X=0)$ This is equal to $1-(0.1)(0.3)(0.16)(0.25) = 0.9988$

Calculating the compliment of the chance of failing in all trials is undoubtedly the 'correct' approach to this problem, as per Siong Thye Goh's answer.

The same result can be calculated by accumulating the additional chance of success from each trial, given that the trial is needed, like this:

Chance this    Chance of success  Additional      Cumulative chance
trial needed   in this trial      success from    chance of success
                                  this trial      in any trial
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A = 1.0 - D'     B (input data)     C = AB         D = C + D'
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                                                   0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 
      1           0.9              0.9             0.9
    0.1           0.7              0.07            0.97
   0.03           0.84             0.0252          0.9952
 0.0048           0.75             0.0036          0.9988