In exothermic reactions, the rate constant decreases on increasing the temperature. When we plot the graph between ln k and 1/T, a positive slope is obtained which gives negative activation energy. Is that valid to calculate the activation energy from Arrhenius plot in this case?
The Arrhenius plot is a model to calculate coefficients of a reaction. So yes you can use it to model exothermic reactions and you will get a negative 'activation energy'. You can go on and use the calculated values to model your reaction.
Be wary that exothermic reactions, or reactions that have a lower reaction rate at higher temperatures saying the 'activation energy is negative' makes no sense. In essence a reaction will not happen by taking away energy, it will require always some energy in order to break bonds and form new molecules. In reality for endothermic reactions the activation energy will be close to 0 and by increasing the temperature you reduce the reaction speed because of diffusion limitations or the kinetic energy of the molecules being to high resulting in less effective collisions to form new molecules.
https://en.wikipedia.org/wiki/Activation_energy