In a competitive market, is it possible to know the change (increase/decrease) in number of firms in the Long run with a positive shift in demand for increasing costs case?
Yes, it is possible. In the long run, firms enter until they break even. Suppose firms are symmetric. Then for each firm the break even condition is that the average costs equal the price. This is because the price is equal to the average revenue. The average revenue is given by $px/x=p$ where $p$ is price and $x$ is quantity. If revenue equals costs on average, then there are no profits left to be made and this is the equilibrium number of firms.
Further, in a competitive market, price equals marginal costs. Hence, the break even condition there is also price equals marginal costs. Alternatively, in this case the condition is Average Costs equal Marginal Costs. From this condition you can find the break even quantity. From there find the break even price, by setting price=marginal costs. Then plug in the break even price and quantity into the demand function to determine how many firms must enter until the market equilibrium price and quantity are equal to the break even price and quantity. The result is the long run equilibrium number of firms.
Note that the question is only reasonable in the presence of fixed costs. If there are no fixed costs, the long run number of firms is infinite and we have a trivial answer. However, the treatment of this topic in textbooks as well as academic papers alwa assumes the presence of fixed costs.