Let $X$ be a (not necessarily metrizable) Hausdorff compact space of covering dimension = 1. 



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> Is it possible to express $X$ as a filtering projective limit of finite graphs? 
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Here finite graphs means topological realizations of finite graphs, with (surjective?) continuous maps between them.



One-dimensional compacta as projective limits
References to Engelking's Dimension Theory (1978) ISBN 0-444-85176-3.
The answer is yes for compact metrizable spaces, see Section 1.13. 
In general it is no in general, see Example 3.3.8 (Lokucievskii's example of a compact space $X$ with $\dim X=1$ and $\mathop{\mathrm{ind}}X=2$). 


