To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$
$$ \eta(s) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^s}} $$
when the real part of $s$ is larger than zero $\Re(s)>0$. Take the partial summations $\Phi: \mathbb{N}\times \mathbb{C}\rightarrow \mathbb{R}$ defined by the rule
$$ \Phi(m,s) = \sum_{n=1}^{m}{\dfrac{(-1)^{n-1}}{n^s}}. $$
To separate the real and imaginary components of $\Phi(m,s)$ let $s = \sigma + it$ with $\sigma = \Re(s)$ and $t = \Im(s)$. Then one can write $\Phi(m, s)$ as
$$ \Phi(m, s) = \sum_{n=1}^{m} \frac{(-1)^{n-1}}{n^{\sigma}} \left( \cos(t \ln(n)) - i \sin(t \ln(n)) \right). $$
The real and imaginary parts of $\Phi(m, s)$ can therefore be expressed as
$$ \Re(\Phi(m, s)) = \sum_{n=1}^{m} \frac{(-1)^{n-1}}{n^{\sigma}} \cos(t \ln(n)), $$
$$ \Im(\Phi(m, s)) = -\sum_{n=1}^{m} \frac{(-1)^{n-1}}{n^{\sigma}} \sin(t \ln(n)). $$
I noticed the distance between $\Re(\eta(s))$ and $\Re(\Phi(m,s))$ defined as the function $$f(k,s):= |\Re(\eta(s)-\Re(\Phi(k,s))|$$ appears to be minimized in the half way point when the trig functions change from positive to negative values. That is $f(k,s)$ is approximately zero when $k = \dfrac{m_i+m_{i+1}}{2}$ for some $m_i,m_{i+1}$ inside $\mathbb{M}$. I will state it more thoroughly as a theorem.
$\textbf{Theorem 1}$ Take $s = \delta + it$ with $0<\delta<1$ and consider the sets $$\mathbb{M}_1(t) = \{i\in \mathbb{N}:~ \cos(t\ln(i))<0~~~~ \text{and}~~~~\cos(t\ln(i+1))>0\}$$
$$\mathbb{M}_2(t) = \{i\in \mathbb{N}:~ \cos(t\ln(i))>0~~~~ \text{and}~~~~\cos(t\ln(i+1))<0\}$$
and then define $$\mathbb{M} = \mathbb{M}_1\cup \mathbb{M}_2$$ with $m_i\le m_{i+1}$ for $i\in \mathbb{N}$ (the set $\mathbb{M}_1\cup\mathbb{M}_2$ well ordered from least to the greatest element). For $(i,s, m_i, m_{i+1})\in \mathbb{N}\times\mathbb{C}\times \mathbb{M}\times\mathbb{M}$ with $0<\Re(s)<1$ we have $$\Re(\Phi(m_i,s))\le \Re(\eta(s))\le \Re(\Phi(m_{i+1},s))$$ or $$\Re(\Phi(m_i,s))\ge \Re(\eta(s))\ge\Re(\Phi(m_{i+1},s)).$$
similarly the imaginary part is bounded by the partial sums (when the trig changes sign) i.e
$\textbf{Theorem 2}$ Take $s = \delta + it$ with $0<\delta<1$ and consider the sets $$\mathbb{M}_1(t) = \{i\in \mathbb{N}:~ \sin(t\ln(i))<0~~~~ \text{and}~~~~\sin(t\ln(i+1))>0\}$$
$$\mathbb{M}_2(t) = \{i\in \mathbb{N}:~ \sin(t\ln(i))>0~~~~ \text{and}~~~~\sin(t\ln(i+1))<0\}$$
and then define $$\mathbb{M} = \mathbb{M}_1\cup \mathbb{M}_2$$ with $m_i\le m_{i+1}$ for $i\in \mathbb{N}$ (the set $\mathbb{M}_1\cup\mathbb{M}_2$ well ordered from least to the greatest element) For $(i,s, m_i, m_{i+1})\in \mathbb{N}\times\mathbb{C}\times \mathbb{M}\times\mathbb{M}$ with $0<\Re(s)<1$ we have $$\Im(\Phi(m_i,s))\le \Im(\eta(s))\le \Im(\Phi(m_{i+1},s))$$ or $$\Im(\Phi(m_i,s))\ge \Im(\eta(s))\ge\Im(\Phi(m_{i+1},s)).$$
Which lastly, leads me to the theorem
$\textbf{Theorem 3}$ Define $\mathbb{M} = \mathbb{M}_1\cup\mathbb{M}_2$ with $$\mathbb{M}_1(t) = \{i\in \mathbb{N}:~ \cos(t\ln(i))<0~~~~ \text{and}~~~~\cos(t\ln(i+1))>0\}$$ and $$\mathbb{M}_2(t) = \{i\in \mathbb{N}:~ \cos(t\ln(i))>0~~~~ \text{and}~~~~\cos(t\ln(i+1))<0\}$$
and Define $\mathbb{M}^\prime = \mathbb{M}_1^\prime\cup\mathbb{M}_2^\prime$ with $$\mathbb{M}_1(t)^\prime = \{i\in \mathbb{N}:~ \sin(t\ln(i))<0~~~~ \text{and}~~~~\sin(t\ln(i+1))>0\}$$ and $$\mathbb{M}_2(t)^\prime = \{i\in \mathbb{N}:~ \sin(t\ln(i))>0~~~~ \text{and}~~~~\sin(t\ln(i+1))<0\}$$
Let $(i,l)\in \mathbb{N}\times \mathbb{N}$ and consider any two adjacent elements in $m_i, m_{i+1}\in \mathbb{M}$, and $m_l^\prime, m_{l+1}^\prime\in \mathbb{M}^\prime$ that are in order. Set $$j = \lfloor(\dfrac{m_{i+1}-m_{i}}{2})\rfloor $$ then $$\Re(\eta(s)) \approx \Re(\Phi(j,s))$$ we need to ensure $j$ is an integer so we have to take the floor function. Similarly, set $$j^\prime = \lfloor(\dfrac{m_{l+1}^\prime-m_{l}^\prime}{2})\rfloor $$ then $$\Im(\eta(s)) \approx \Im(\Phi(j^\prime,s)).$$
As example, take $j = \lfloor \dfrac{6606-5505}{2}\rfloor$ and let $s = 0.32+ 15i$. My graph suggest $ |\Re(\eta(s))-\Re(\Phi(j,s))|< |\Re(\eta(s))-\Re(\Phi(k,s))| $ for any $k\in (5505,6606)$.
